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<section class="center" data-background-image="intro-to-r-for-cs1_files/assets/milky-way.jpg">
<h1 class="title">Introduction to R via CS1</h1>
<h2 class="author">Alex Garbiak</h2>
<h3 class="date">24 August, 2020</h3>
</section>
<section id="why-are-we-here" class="title-slide slide level1 white center" data-background="intro-to-r-for-cs1_files/assets/abstract.jpg"><h1>Why are we here?</h1><ul>
<li class="fragment">To pass the <strong>CS1</strong> <em>(Part B)</em> exam 👍</li>
<li class="fragment">To be persuaded that <span class="math inline">\(R\)</span> is great for actuarial work 👀 … maybe next time.</li>
</ul></section>
<section><section id="the-basics" class="title-slide slide level1 white center" data-background="intro-to-r-for-cs1_files/assets/coding.jpg"><h1>The Basics <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg></h1><p><span class="footnote">Yes … it’s <code>HTML</code>, not <code>R</code>😏</span></p></section><section id="background" class="slide level2">
<h2>Background 📚</h2>
<p>This presentation is to help <a href="https://www.actuaries.org.uk/">IFoA</a> actuaries tackle <strong>Part B</strong> of the <strong>CS1</strong> exam.</p>
<ul>
<li class="fragment">Some <strong>prerequisites</strong>:
<ul>
<li class="fragment">Understand the <strong>CS1</strong> <a href="https://www.actuaries.org.uk/studying/curriculum/actuarial-statistics">Actuarial Statistics</a> material</li>
<li class="fragment">Install <a href="https://cran.rstudio.com/"><code>R</code></a></li>
<li class="fragment">Install <a href="https://rstudio.com/products/rstudio/download/#download">RStudio</a></li>
</ul></li>
</ul>
</section><section id="rstudio-layout" class="slide level2">
<h2>RStudio layout 🌍</h2>
<p><img data-src="intro-to-r-for-cs1_files/assets/rstudio-layout.png" alt="RStudio layout discussion" /></p>
</section><section id="r-loves-vectors-️" class="slide level2">
<h2>R loves vectors ❤️</h2>
<p>R prefers <strong>vectorised operations</strong> (<em>over concepts like for loops</em>)</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1"></a><span class="co"># This is the syntax for comments in R</span></span>
<span id="cb1-2"><a href="#cb1-2"></a>(<span class="dv">1</span><span class="op">:</span><span class="dv">10</span>) <span class="op">+</span><span class="st"> </span><span class="dv">2</span> <span class="co"># Notice how we add element-wise in R</span></span></code></pre></div>
<pre><code>## [1] 3 4 5 6 7 8 9 10 11 12</code></pre>
</section><section id="r-also-loves-variables" class="slide level2 leftAligned">
<h2>R also loves variables 💘</h2>
<p>We <strong>assign</strong> values to <strong>variables</strong> using the <code><-</code> <em>(“assignment”)</em> operator</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1"></a>x <-<span class="st"> </span><span class="dv">1</span><span class="op">:</span><span class="dv">10</span></span>
<span id="cb3-2"><a href="#cb3-2"></a>y <-<span class="st"> </span>x <span class="op">+</span><span class="st"> </span><span class="dv">2</span></span>
<span id="cb3-3"><a href="#cb3-3"></a>x <-<span class="st"> </span>x <span class="op">+</span><span class="st"> </span>x <span class="co"># Notice that we can re-assign values to variables</span></span>
<span id="cb3-4"><a href="#cb3-4"></a>z <-<span class="st"> </span>x <span class="op">+</span><span class="st"> </span><span class="dv">2</span></span>
<span id="cb3-5"><a href="#cb3-5"></a>y</span>
<span id="cb3-6"><a href="#cb3-6"></a><span class="co">## [1] 3 4 5 6 7 8 9 10 11 12</span></span>
<span id="cb3-7"><a href="#cb3-7"></a>z</span>
<span id="cb3-8"><a href="#cb3-8"></a><span class="co">## [1] 4 6 8 10 12 14 16 18 20 22</span></span></code></pre></div>
<p>Even though <span class="math inline">\(z\)</span> is assigned the same way as we assigned <span class="math inline">\(y\)</span>, note that <span class="math inline">\(y \neq z\)</span> so execution order matters in <span class="math inline">\(R\)</span></p>
</section><section id="add-power-with-functions" class="slide level2">
<h2>Add power with functions 🚀</h2>
<p>We now add <strong>functions</strong> to the <span class="math inline">\(R\)</span> code which has the form <code class="sourceCode r"><span class="kw">function_name</span>(<span class="dt">arguments =</span> <span class="st">"values"</span>, ...)</code></p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1"></a><span class="co"># Combine function, used often to create vectors:</span></span>
<span id="cb4-2"><a href="#cb4-2"></a>x <-<span class="st"> </span><span class="kw">c</span>(<span class="dv">1</span><span class="op">:</span><span class="dv">3</span>, <span class="dv">6</span><span class="op">:</span><span class="dv">20</span>, <span class="dv">21</span><span class="op">:</span><span class="dv">42</span>) </span>
<span id="cb4-3"><a href="#cb4-3"></a><span class="co"># Another function with arguments:</span></span>
<span id="cb4-4"><a href="#cb4-4"></a>y <-<span class="st"> </span><span class="kw">sample</span>(x, <span class="dt">size =</span> <span class="dv">3</span>)</span>
<span id="cb4-5"><a href="#cb4-5"></a>y</span></code></pre></div>
<pre><code>## [1] 21 37 23</code></pre>
</section><section id="matrices" class="slide level2">
<h2>Matrices 🏛</h2>
<p>Let’s create a <strong>matrix</strong> in <span class="math inline">\(R\)</span></p>
<p><em>Note:</em> <strong>Matrix multiplication</strong> requires the <code>%*%</code> syntax</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1"></a>first_matrix <-<span class="st"> </span><span class="kw">matrix</span>(<span class="dv">1</span><span class="op">:</span><span class="dv">9</span>, <span class="dt">byrow =</span> <span class="ot">TRUE</span>, <span class="dt">nrow =</span> <span class="dv">3</span>)</span>
<span id="cb6-2"><a href="#cb6-2"></a>first_matrix <span class="op">%*%</span><span class="st"> </span>first_matrix</span></code></pre></div>
<pre><code>## [,1] [,2] [,3]
## [1,] 30 36 42
## [2,] 66 81 96
## [3,] 102 126 150</code></pre>
</section><section id="dataframes" class="slide level2 leftAligned">
<h2>Dataframes 🏭</h2>
<p>A <code>data.frame</code> is a very popular data structure used in <span class="math inline">\(R\)</span>. Each input variable has to have the same length but can be of different types (<em>strings, integers, booleans, etc.</em>).</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1"></a><span class="co"># Input vectors for the data.frame</span></span>
<span id="cb8-2"><a href="#cb8-2"></a>name <-<span class="st"> </span><span class="kw">c</span>(<span class="st">"Mercury"</span>, <span class="st">"Venus"</span>, <span class="st">"Earth"</span>, <span class="st">"Mars"</span>, <span class="st">"Jupiter"</span>, <span class="st">"Saturn"</span>, <span class="st">"Uranus"</span>, <span class="st">"Neptune"</span>)</span>
<span id="cb8-3"><a href="#cb8-3"></a>surface_gravity <-<span class="st"> </span><span class="kw">c</span>(<span class="fl">0.38</span>, <span class="fl">0.904</span>, <span class="dv">1</span>, <span class="fl">0.3794</span>, <span class="fl">2.528</span>, <span class="fl">1.065</span>, <span class="fl">0.886</span>, <span class="fl">1.14</span>)</span>
<span id="cb8-4"><a href="#cb8-4"></a></span>
<span id="cb8-5"><a href="#cb8-5"></a><span class="co"># Create a data.frame from the vectors</span></span>
<span id="cb8-6"><a href="#cb8-6"></a>solar_system <-<span class="st"> </span><span class="kw">data.frame</span>(name, surface_gravity)</span>
<span id="cb8-7"><a href="#cb8-7"></a><span class="kw">str</span>(solar_system)</span></code></pre></div>
<pre><code>## 'data.frame': 8 obs. of 2 variables:
## $ name : chr "Mercury" "Venus" "Earth" "Mars" ...
## $ surface_gravity: num 0.38 0.904 1 0.379 2.528 ...</code></pre>
</section><section id="logical-expressions-️" class="slide level2">
<h2>Logical expressions ⚗️</h2>
<p>R has built in logic expressions:</p>
<table>
<thead>
<tr class="header">
<th>Operator</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>< (<=)</td>
<td>less than (or equal to)</td>
</tr>
<tr class="even">
<td>> (>=)</td>
<td>greater than (or equal to)</td>
</tr>
<tr class="odd">
<td>==</td>
<td>exactly equal to</td>
</tr>
<tr class="even">
<td>!</td>
<td>NOT</td>
</tr>
<tr class="odd">
<td>&</td>
<td>AND (<em>element-wise</em>)</td>
</tr>
<tr class="even">
<td>|</td>
<td>OR (<em>element-wise</em>)</td>
</tr>
<tr class="odd">
<td>!=</td>
<td>not equal to</td>
</tr>
</tbody>
</table>
</section><section id="filtering-and-subsetting" class="slide level2 leftAligned">
<h2>Filtering and Subsetting 🔬</h2>
<p>We can use logical expressions to effectively filter data</p>
<p>Here we <strong>subset</strong> the data using the <code class="sourceCode r">[...]</code> syntax</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1"></a>x <-<span class="st"> </span><span class="dv">1</span><span class="op">:</span><span class="dv">10</span></span>
<span id="cb10-2"><a href="#cb10-2"></a>x[x <span class="op">!=</span><span class="st"> </span><span class="dv">5</span> <span class="op">&</span><span class="st"> </span>x <span class="op"><</span><span class="st"> </span><span class="dv">7</span>]</span></code></pre></div>
<pre><code>## [1] 1 2 3 4 6</code></pre>
</section><section id="selecting-objects" class="slide level2 leftAligned">
<h2>Selecting objects 🔬</h2>
<p>We can select objects using the <strong>$</strong> symbol - see <code class="sourceCode r">?Extract</code> for more help here</p>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1"></a><span class="co">#data.frame[rows to select, columns to select]</span></span>
<span id="cb12-2"><a href="#cb12-2"></a>solar_system[solar_system<span class="op">$</span>name <span class="op">==</span><span class="st"> "Jupiter"</span>, <span class="kw">c</span>(<span class="dv">1</span><span class="op">:</span><span class="dv">2</span>)]</span></code></pre></div>
<pre><code>## name surface_gravity
## 5 Jupiter 2.528</code></pre>
</section><section id="expand-r-with-packages" class="slide level2 leftAligned">
<h2>Expand R with packages 📦</h2>
<p>We can extend <span class="math inline">\(R\)</span>’s functionality by loading <strong>packages</strong>:</p>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1"></a><span class="co"># Load the ggplot2 package</span></span>
<span id="cb14-2"><a href="#cb14-2"></a><span class="kw">library</span>(ggplot2)</span></code></pre></div>
<ul>
<li class="fragment">Did you get an error from <span class="math inline">\(R\)</span> trying this?</li>
<li class="fragment">To load packages they need to be <strong>installed</strong> first:</li>
<li class="fragment"><code class="sourceCode r"><span class="kw">install.packages</span>(<span class="st">"ggplot2"</span>)</code></li>
</ul>
</section><section id="importing-data" class="slide level2 leftAligned">
<h2>Importing data 📃</h2>
<p>We’ll focus on exam likely candidates with <code>.csv</code> and <code>.RData</code> file imports</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1"></a><span class="co"># Import csv files with read.csv() function:</span></span>
<span id="cb15-2"><a href="#cb15-2"></a>csv_data <-<span class="st"> </span></span>
<span id="cb15-3"><a href="#cb15-3"></a><span class="st"> </span><span class="kw">read.csv</span>(<span class="st">"intro-to-r-for-cs1_files/data/USTreasury_1YR.csv"</span>)</span>
<span id="cb15-4"><a href="#cb15-4"></a></span>
<span id="cb15-5"><a href="#cb15-5"></a><span class="co"># For RData files, we use the load() function:</span></span>
<span id="cb15-6"><a href="#cb15-6"></a><span class="kw">load</span>(<span class="st">"intro-to-r-for-cs1_files/data/USTreasury_10YR.RData"</span>)</span></code></pre></div>
<p><code>.txt</code> files can be loaded using the <code class="sourceCode r"><span class="kw">read.table</span>()</code> function</p>
</section><section id="random-variables" class="slide level2">
<h2>Random Variables 🎰</h2>
<p><span class="math inline">\(R\)</span> was designed to be used for statistical computing - so it handles <strong>randomness</strong> well!</p>
<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1"></a><span class="kw">set.seed</span>(<span class="dv">42</span>) <span class="co"># Fixes result</span></span>
<span id="cb16-2"><a href="#cb16-2"></a>die_throws <-<span class="st"> </span><span class="kw">sample</span>(<span class="dv">1</span><span class="op">:</span><span class="dv">6</span>, <span class="dv">10000</span>, <span class="dt">replace =</span> <span class="ot">TRUE</span>)</span>
<span id="cb16-3"><a href="#cb16-3"></a><span class="kw">mean</span>(die_throws)</span></code></pre></div>
<pre><code>## [1] 3.4627</code></pre>
</section><section id="probability-distributions" class="slide level2">
<h2>Probability Distributions 🎰</h2>
<ul>
<li class="fragment"><code>R</code> has in-built functions for probability distributions:
<ul>
<li class="fragment"><strong>d</strong><em><distribution-name></em> <span class="math inline">\(:=\)</span> <strong>d</strong>ensity (PDF), <em>i.e.</em> <span class="math inline">\(f_X(x)\)</span></li>
<li class="fragment"><strong>p</strong><em><distribution-name></em> <span class="math inline">\(:=\)</span> <strong>p</strong>robability distribution cumulative function (CDF), <em>i.e.</em> <span class="math inline">\(F_X(x) =\boldsymbol{P}(X \leq x)\)</span></li>
<li class="fragment"><strong>q</strong><em><distribution-name></em> <span class="math inline">\(:=\)</span> <strong>q</strong>uantile function, <em>i.e.</em> return <span class="math inline">\(x\)</span> such that <span class="math inline">\(\boldsymbol{P}(X \leq x) = p\)</span></li>
<li class="fragment"><strong>r</strong><em><distribution-name></em> <span class="math inline">\(:=\)</span> <strong>r</strong>andom deviates, <em>i.e.</em> (<em>psuedo</em>) random number generator for a given distribution</li>
<li class="fragment"><em>Where <distribution-name></em> <span class="math inline">\(=\)</span> Normal, uniform, lognormal, Student’s t, Poisson, binormal, Weibull … see <code class="sourceCode r">?<span class="kw">distributions</span>()</code> for more information</li>
</ul></li>
</ul>
</section><section id="probability-distributions-examples" class="slide level2">
<h2>Probability Distributions Examples 🎰</h2>
<table>
<colgroup>
<col style="width: 50%" />
<col style="width: 50%" />
</colgroup>
<thead>
<tr class="header">
<th>R Code</th>
<th>Definition</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code class="sourceCode r"><span class="kw">rnorm</span>(<span class="dv">1</span>)</code></td>
<td>Generates <span class="math inline">\(x_1\)</span> where <span class="math inline">\(X \sim \mathcal{N}(0,\,1)\)</span></td>
</tr>
<tr class="even">
<td><code class="sourceCode r"><span class="kw">rnorm</span>(y, <span class="dt">mean=</span><span class="dv">10</span>, <span class="dt">sd=</span><span class="dv">2</span>)</code></td>
<td>Generates <span class="math inline">\(\{y_1,\,y_2,\,\dots\}\)</span> with <span class="math inline">\(Y \sim \mathcal{N}(10,\,2^2)\)</span></td>
</tr>
<tr class="odd">
<td><code class="sourceCode r"><span class="kw">runif</span>(<span class="dv">3</span>, <span class="dt">min=</span><span class="dv">5</span>, <span class="dt">max=</span><span class="dv">10</span>)</code></td>
<td>Generates <span class="math inline">\(\{z_1,\,z_2,\,z_3\}\)</span> where <span class="math inline">\(Z \sim \mathcal{U}(5,\,10)\)</span></td>
</tr>
<tr class="even">
<td><code class="sourceCode r"><span class="kw">dbinom</span>(<span class="dv">4</span>, <span class="dt">size=</span><span class="dv">5</span>, <span class="dt">prob=</span><span class="fl">0.5</span>)</code></td>
<td>Computes <span class="math inline">\(\boldsymbol{P}(X = 4)\)</span> where <span class="math inline">\(X \sim \mathcal{Bin}(5,\,0.5)\)</span></td>
</tr>
<tr class="odd">
<td><code class="sourceCode r"><span class="kw">pgamma</span>(<span class="fl">0.2</span>, <span class="dt">shape=</span><span class="dv">2</span>, <span class="dt">rate=</span><span class="dv">2</span>)</code><br><span class="footnote">*See footnote</span></td>
<td>Computes <span class="math inline">\(F_Y(0.2)\)</span> where <span class="math inline">\(Y \sim \mathcal{\Gamma}(2,\,2)\)</span>, i.e. <span class="math inline">\(\boldsymbol{P}(Y\leq 0.2)\)</span></td>
</tr>
<tr class="even">
<td><code class="sourceCode r"><span class="kw">qexp</span>(<span class="fl">0.5</span>, <span class="dt">rate =</span> <span class="dv">2</span>)</code></td>
<td>Determines smallest value of <span class="math inline">\(z\)</span> for <span class="math inline">\(\boldsymbol{P}(Z \leq z) = 0.5\)</span> where <span class="math inline">\(Z \sim Exp(2)\)</span></td>
</tr>
</tbody>
</table>
<p><span class="footnote">*This is how the <span class="math inline">\(\Gamma\)</span> distribution is parametrised in “<em>FORMULAE AND TABLES for EXAMINATIONS of THE INSTITUTE and FACULTY OF ACTUARIES</em>”</span></p>
</section></section>
<section><section id="data-analysis" class="title-slide slide level1" data-background="intro-to-r-for-cs1_files/assets/lagoon-nebula.jpg"><h1>Data Analysis <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg></h1></section><section id="exploratory-data-analysis" class="slide level2">
<h2>Exploratory Data Analysis 🔬</h2>
<p><code>R</code> is great for calculating “<em>suitable summary statistics</em>”</p>
<div class="sourceCode" id="cb18"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1"></a>x <-<span class="st"> </span><span class="kw">sample</span>(<span class="dv">1</span><span class="op">:</span><span class="dv">100</span>, <span class="dv">10</span>)</span>
<span id="cb18-2"><a href="#cb18-2"></a><span class="kw">summary</span>(x)</span></code></pre></div>
<pre><code>## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 12.00 25.25 45.50 51.50 82.75 93.00</code></pre>
<div class="sourceCode" id="cb20"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1"></a><span class="kw">sd</span>(x) <span class="co"># Or: sqrt(var(x)) </span></span></code></pre></div>
<pre><code>## [1] 31.80234</code></pre>
</section><section id="calculating-correlations" class="slide level2 leftAligned">
<h2>Calculating correlations 📓</h2>
<p>Let’s calculate the “<em>Pearson’s, Spearman’s and Kendall’s measures of correlation for bivariate data</em>” using the <code class="sourceCode r"><span class="kw">cor</span>(x, y, <span class="dt">method =</span> <span class="st">"method-name"</span>)</code> function*:</p>
<div class="sourceCode" id="cb22"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1"></a><span class="co"># Note: Data was loaded in the "Importing Data" section*</span></span>
<span id="cb22-2"><a href="#cb22-2"></a><span class="kw">cor</span>(</span>
<span id="cb22-3"><a href="#cb22-3"></a> <span class="dt">x =</span> csv_data<span class="op">$</span>WGS1YR, </span>
<span id="cb22-4"><a href="#cb22-4"></a> <span class="dt">y =</span> USTreasury_10YR<span class="op">$</span>WGS10YR, </span>
<span id="cb22-5"><a href="#cb22-5"></a> <span class="dt">method =</span> <span class="st">"pearson"</span></span>
<span id="cb22-6"><a href="#cb22-6"></a> )</span></code></pre></div>
<pre><code>## [1] 0.2548368</code></pre>
<p><span class="footnote">*Using the US Treasury data we loaded <a href="#/importing-data">here</a>.</span></p>
</section></section>
<section><section id="statistical-inference" class="title-slide slide level1 center" data-background="intro-to-r-for-cs1_files/assets/stats.jpg"><h1>Statistical inference <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg></h1></section><section id="population-mean-confidence-intervals" class="slide level2 leftAligned">
<h2>Population mean confidence intervals 🔭</h2>
<p>Let’s calculate confidence intervals for “<em>the mean … of a normal distribution</em>” using Student’s t-Test using the <code class="sourceCode r"><span class="kw">t.test</span>()</code> function:</p>
<div class="sourceCode" id="cb24"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1"></a><span class="kw">set.seed</span>(<span class="dv">42</span>) <span class="co"># To fix result</span></span>
<span id="cb24-2"><a href="#cb24-2"></a><span class="co"># We create x_1, x_2, ... x_100 with X ~ N(0,1)</span></span>
<span id="cb24-3"><a href="#cb24-3"></a>x <-<span class="st"> </span><span class="kw">rnorm</span>(<span class="dv">100</span>)</span>
<span id="cb24-4"><a href="#cb24-4"></a></span>
<span id="cb24-5"><a href="#cb24-5"></a><span class="co"># 99% CI for population mean:</span></span>
<span id="cb24-6"><a href="#cb24-6"></a><span class="kw">as.vector</span>(</span>
<span id="cb24-7"><a href="#cb24-7"></a> <span class="kw">t.test</span>(x, <span class="dt">conf.level =</span> <span class="fl">0.99</span>)<span class="op">$</span>conf.int</span>
<span id="cb24-8"><a href="#cb24-8"></a> )</span></code></pre></div>
<pre><code>## [1] -0.2409877 0.3060174</code></pre>
</section><section id="population-variance-confidence-intervals" class="slide level2 leftAligned">
<h2>Population variance confidence intervals 🔭</h2>
<p>We can also calculate confidence intervals for “<em>the variance of a normal distribution</em>” using the <span class="math inline">\(\chi^2\)</span> distribution:</p>
<div class="sourceCode" id="cb26"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1"></a><span class="kw">set.seed</span>(<span class="dv">42</span>) <span class="co"># To fix result</span></span>
<span id="cb26-2"><a href="#cb26-2"></a><span class="co"># We create x_1, x_2, ... x_100 with X ~ N(0,1)</span></span>
<span id="cb26-3"><a href="#cb26-3"></a>x <-<span class="st"> </span><span class="kw">rnorm</span>(<span class="dv">100</span>)</span>
<span id="cb26-4"><a href="#cb26-4"></a></span>
<span id="cb26-5"><a href="#cb26-5"></a>degrees_freedom <-<span class="st"> </span><span class="kw">length</span>(x) <span class="op">-</span><span class="st"> </span><span class="dv">1</span></span>
<span id="cb26-6"><a href="#cb26-6"></a>sample_var_x <-<span class="st"> </span><span class="kw">var</span>(x)</span>
<span id="cb26-7"><a href="#cb26-7"></a><span class="co"># 95% CI for population variance:</span></span>
<span id="cb26-8"><a href="#cb26-8"></a><span class="kw">c</span>(</span>
<span id="cb26-9"><a href="#cb26-9"></a> degrees_freedom <span class="op">*</span><span class="st"> </span>sample_var_x <span class="op">/</span><span class="st"> </span><span class="kw">qchisq</span>(<span class="fl">0.975</span>, degrees_freedom),</span>
<span id="cb26-10"><a href="#cb26-10"></a> degrees_freedom <span class="op">*</span><span class="st"> </span>sample_var_x <span class="op">/</span><span class="st"> </span><span class="kw">qchisq</span>(<span class="fl">0.025</span>, degrees_freedom)</span>
<span id="cb26-11"><a href="#cb26-11"></a>)</span></code></pre></div>
<pre><code>## [1] 0.8359784 1.4634191</code></pre>
</section><section id="simple-linear-regression" class="slide level2 leftAligned">
<h2>Simple linear regression 📡</h2>
<p>Let’s “<em>fit a simple linear regression model to a data set</em>” using the <code class="sourceCode r"><span class="kw">lm</span>(formula)</code> function and the symbolic syntax <code class="sourceCode r">predicted <span class="op">~</span><span class="st"> </span><span class="kw">explanatory</span>(s)</code></p>
<div class="sourceCode" id="cb28"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1"></a><span class="co"># Note: "women" is a built-in dataset in R</span></span>
<span id="cb28-2"><a href="#cb28-2"></a>simple_linear_model <-<span class="st"> </span><span class="kw">lm</span>(weight <span class="op">~</span><span class="st"> </span>height, <span class="dt">data =</span> women)</span>
<span id="cb28-3"><a href="#cb28-3"></a><span class="kw">summary</span>(simple_linear_model)<span class="op">$</span>coefficients</span></code></pre></div>
<pre><code>## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -87.51667 5.9369440 -14.74103 1.711082e-09
## height 3.45000 0.0911365 37.85531 1.090973e-14</code></pre>
<p>We have a model for weight as:</p>
<p>weight = intercept<span class="math inline">\(_{(-87.52)}\)</span> + <span class="math inline">\(\beta _{(3.45)}\)</span> * height</p>
</section><section id="multiple-linear-regression" class="slide level2 leftAligned">
<h2>Multiple linear regression 🚀</h2>
<p>Let’s expand this and “<em>fit a multiple linear regression model to a data set</em>” using the <code class="sourceCode r"><span class="kw">lm</span>()</code> function again, a <code>data.frame</code> as our data and the symbolic notation:</p>
<p><span class="math inline">\(\hat{Y} = \beta_0 + \beta_1X_1 + \beta_2X_2 + \,_{\dots}\, + \beta_nX_n\)</span></p>
<div class="sourceCode" id="cb30"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1"></a><span class="co"># NOT EVALUATED</span></span>
<span id="cb30-2"><a href="#cb30-2"></a>better_model <-<span class="st"> </span><span class="kw">lm</span>(y <span class="op">~</span><span class="st"> </span>x1 <span class="op">+</span><span class="st"> </span>x2 <span class="op">+</span><span class="st"> </span>x3, <span class="dt">data =</span> data_set)</span>
<span id="cb30-3"><a href="#cb30-3"></a><span class="kw">summary</span>(multiple_linear_model) <span class="co"># Provides full results of the lm</span></span>
<span id="cb30-4"><a href="#cb30-4"></a></span>
<span id="cb30-5"><a href="#cb30-5"></a><span class="co"># Residual standard error:</span></span>
<span id="cb30-6"><a href="#cb30-6"></a><span class="kw">sigma</span>(better_model) <span class="op">/</span><span class="st"> </span><span class="kw">mean</span>(data_set<span class="op">$</span>y)</span>
<span id="cb30-7"><a href="#cb30-7"></a></span>
<span id="cb30-8"><a href="#cb30-8"></a><span class="co"># Compare models:</span></span>
<span id="cb30-9"><a href="#cb30-9"></a>new_model <-<span class="st"> </span><span class="kw">lm</span>(y <span class="op">~</span><span class="st"> </span>x1 <span class="op">+</span><span class="st"> </span>x2)</span>
<span id="cb30-10"><a href="#cb30-10"></a><span class="kw">anova</span>(better_model, new_model)</span></code></pre></div>
</section><section id="generalised-linear-models" class="slide level2 leftAligned">
<h2>Generalised linear models 🌌</h2>
<p>To “<em>fit a generalised linear model to a data set</em>” in <span class="math inline">\(R\)</span> we make use of the <code class="sourceCode r"><span class="kw">glm</span>()</code> function. Here we generalise:</p>
<ul>
<li class="fragment">Predicted variable <span class="math inline">\(Y\)</span> follows a exponential <strong>family</strong> distribution</li>
<li class="fragment">Explanatory variables form a <strong>linear predictor</strong> <span class="math inline">\(\beta_0 + \beta_1X_1, \, \beta_2X_2, \, \dots, \, \beta_nX_n\)</span></li>
<li class="fragment">We link <span class="math inline">\(Y\)</span> and the linear predictor using a <strong>link function</strong></li>
</ul>
<div class="sourceCode" id="cb31"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1"></a><span class="co"># NOT EVALUATED</span></span>
<span id="cb31-2"><a href="#cb31-2"></a>glm_model <-<span class="st"> </span></span>
<span id="cb31-3"><a href="#cb31-3"></a><span class="st"> </span><span class="kw">glm</span>(<span class="dt">formula =</span> y <span class="op">~</span><span class="st"> </span>x1 <span class="op">+</span><span class="st"> </span>x2 <span class="op">+</span><span class="st"> </span>x3,</span>
<span id="cb31-4"><a href="#cb31-4"></a> <span class="dt">family =</span> <span class="kw">Gamma</span>(<span class="dt">link =</span> <span class="st">"inverse"</span>), </span>
<span id="cb31-5"><a href="#cb31-5"></a> <span class="dt">data =</span> data_set</span>
<span id="cb31-6"><a href="#cb31-6"></a> )</span></code></pre></div>
</section></section>
<section><section id="plotting" class="title-slide slide level1" data-background="intro-to-r-for-cs1_files/assets/nebula.png"><h1>Plotting <svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg></h1></section><section id="base-plotting" class="slide level2 leftAligned">
<h2>Base plotting 📉</h2>
<p>We need to be able to “… <em>undertake exploratory data visualizations</em>”, here we use the <code class="sourceCode r"><span class="kw">plot</span>()</code> function:</p>
<div class="sourceCode" id="cb32"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1"></a><span class="kw">plot</span>(</span>
<span id="cb32-2"><a href="#cb32-2"></a> <span class="dt">x =</span> USTreasury_10YR<span class="op">$</span>DATE, <span class="dt">y =</span> USTreasury_10YR<span class="op">$</span>WGS10YR, <span class="co"># Data</span></span>
<span id="cb32-3"><a href="#cb32-3"></a> <span class="dt">main =</span> <span class="st">"10-Year Treasury Constant Maturity Rate"</span>, <span class="co"># Title</span></span>
<span id="cb32-4"><a href="#cb32-4"></a> <span class="dt">sub =</span> <span class="st">"Source: Board of Governors of the Federal Reserve System (US)"</span>,</span>
<span id="cb32-5"><a href="#cb32-5"></a> <span class="dt">xlab =</span> <span class="st">"Year"</span>, <span class="dt">ylab =</span> <span class="st">"Percent"</span>, <span class="co"># Axis labels</span></span>
<span id="cb32-6"><a href="#cb32-6"></a> <span class="dt">type =</span> <span class="st">"l"</span> <span class="co"># Type of plot - here we use a l(ine) graph</span></span>
<span id="cb32-7"><a href="#cb32-7"></a> )</span></code></pre></div>
</section><section id="scatter-plot" class="slide level2">
<h2>Scatter plot 📉</h2>
<p><img src="intro-to-r-for-cs1_files/figure-revealjs/base-plot-1.png" width="768" /></p>
</section><section id="producing-q-q-plots" class="slide level2 leftAligned">
<h2>Producing Q-Q plots 📉</h2>
<p>We can produce Q-Q plots using the <code class="sourceCode r"><span class="kw">qqnorm</span>()</code> and <code class="sourceCode r"><span class="kw">qqplot</span>()</code> functions. Add a theoretical line using the <code class="sourceCode r"><span class="kw">qqline</span>()</code> function.</p>
<div class="sourceCode" id="cb33"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1"></a><span class="co"># Note: "ToothGrowth" is a built-in dataset in R</span></span>
<span id="cb33-2"><a href="#cb33-2"></a><span class="kw">qqnorm</span>(ToothGrowth<span class="op">$</span>len, </span>
<span id="cb33-3"><a href="#cb33-3"></a> <span class="dt">pch =</span> <span class="dv">15</span>, <span class="co"># plotting symbol</span></span>
<span id="cb33-4"><a href="#cb33-4"></a> <span class="dt">col =</span> <span class="st">"blue"</span>) <span class="co"># colour </span></span>
<span id="cb33-5"><a href="#cb33-5"></a></span>
<span id="cb33-6"><a href="#cb33-6"></a><span class="co"># Add a reference line:</span></span>
<span id="cb33-7"><a href="#cb33-7"></a><span class="kw">qqline</span>(ToothGrowth<span class="op">$</span>len,</span>
<span id="cb33-8"><a href="#cb33-8"></a> <span class="dt">col =</span> <span class="st">"#FF0000"</span>, <span class="co"># can also use RGB</span></span>
<span id="cb33-9"><a href="#cb33-9"></a> <span class="dt">lwd =</span> <span class="dv">3</span>) <span class="co"># line width</span></span></code></pre></div>
</section><section id="q-q-plot" class="slide level2">
<h2>Q-Q plot 📉</h2>
<p><img src="intro-to-r-for-cs1_files/figure-revealjs/plot-qq-1.png" width="768" /></p>
</section><section id="producing-historgrams" class="slide level2">
<h2>Producing historgrams 📉</h2>
<p>Histograms can be generated using the <code class="sourceCode r"><span class="kw">hist</span>()</code> function:</p>
<div class="sourceCode" id="cb34"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb34-1"><a href="#cb34-1"></a><span class="kw">set.seed</span>(<span class="dv">42</span>) <span class="co"># Fix the result</span></span>
<span id="cb34-2"><a href="#cb34-2"></a></span>
<span id="cb34-3"><a href="#cb34-3"></a><span class="co"># Histogram of random draws from the t-distribution</span></span>
<span id="cb34-4"><a href="#cb34-4"></a><span class="kw">hist</span>(<span class="kw">rt</span>(<span class="dv">1000</span>, <span class="dt">df =</span> <span class="dv">10</span>), </span>
<span id="cb34-5"><a href="#cb34-5"></a> <span class="co"># Use breaks argument to (optionally) control the binning</span></span>
<span id="cb34-6"><a href="#cb34-6"></a> <span class="dt">breaks =</span> <span class="kw">seq</span>(<span class="dt">from=</span><span class="op">-</span><span class="dv">5</span>, <span class="dt">to=</span><span class="dv">5</span>, <span class="dt">by=</span><span class="fl">0.5</span>),</span>
<span id="cb34-7"><a href="#cb34-7"></a> <span class="dt">col =</span> <span class="kw">rgb</span>(<span class="dv">173</span>,<span class="dv">216</span>,<span class="dv">230</span>,<span class="dv">80</span>, <span class="dt">max =</span> <span class="dv">255</span>), <span class="co"># Colour</span></span>
<span id="cb34-8"><a href="#cb34-8"></a> <span class="dt">main =</span> <span class="st">"Histogram of random draws from t-distribution"</span>,</span>
<span id="cb34-9"><a href="#cb34-9"></a> <span class="dt">xlab =</span> <span class="st">"Value"</span></span>
<span id="cb34-10"><a href="#cb34-10"></a> )</span>
<span id="cb34-11"><a href="#cb34-11"></a></span>
<span id="cb34-12"><a href="#cb34-12"></a><span class="co"># Add second histogram</span></span>
<span id="cb34-13"><a href="#cb34-13"></a>y <-<span class="st"> </span><span class="kw">rt</span>(<span class="dv">1000</span>, <span class="dt">df =</span> <span class="dv">20</span>)</span>
<span id="cb34-14"><a href="#cb34-14"></a><span class="kw">hist</span>(y, <span class="dt">col =</span> <span class="kw">rgb</span>(<span class="dv">255</span>,<span class="dv">192</span>,<span class="dv">203</span>,<span class="dv">80</span>, <span class="dt">max =</span> <span class="dv">255</span>), <span class="dt">breaks =</span> <span class="dv">21</span>,</span>
<span id="cb34-15"><a href="#cb34-15"></a> <span class="dt">add =</span> <span class="ot">TRUE</span>) <span class="co"># Add onto existing plot</span></span></code></pre></div>
</section><section id="histogram" class="slide level2">
<h2>Histogram 📉</h2>
<p><img src="intro-to-r-for-cs1_files/figure-revealjs/plot-histogram-1.png" width="768" /></p>
</section><section id="plotting-with-ggplot2" class="slide level2 leftAligned">
<h2>Plotting with ggplot2 📈</h2>
<p>Plot with <code class="sourceCode r"><span class="kw">ggplot</span>()</code> in <code class="sourceCode r"><span class="kw">libary</span>(ggplot2)</code>, defining your aesthetics with <code class="sourceCode r"><span class="kw">aes</span>()</code> and the type of chart with the <code class="sourceCode r"><span class="kw">geom_</span>()</code> functions and you might become a <a href="https://bbc.github.io/rcookbook/">data journalist</a></p>
<div class="sourceCode" id="cb35"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1"></a><span class="co"># Join datasets together: csv_data (treating Date format) and USTreasury_10YR</span></span>
<span id="cb35-2"><a href="#cb35-2"></a>yield_data <-<span class="st"> </span><span class="kw">data.frame</span>(<span class="dt">DATE =</span> <span class="kw">as.Date</span>(csv_data<span class="op">$</span>DATE), <span class="dt">yield_rate =</span> csv_data<span class="op">$</span>WGS1YR, <span class="dt">term =</span> <span class="st">"1 year"</span>)</span>
<span id="cb35-3"><a href="#cb35-3"></a>yield_data <-<span class="st"> </span><span class="kw">rbind</span>(yield_data, </span>
<span id="cb35-4"><a href="#cb35-4"></a> <span class="kw">data.frame</span>(<span class="dt">DATE =</span> USTreasury_10YR<span class="op">$</span>DATE, <span class="dt">yield_rate =</span> USTreasury_10YR<span class="op">$</span>WGS10YR, <span class="dt">term =</span> <span class="st">"10 year"</span>))</span>
<span id="cb35-5"><a href="#cb35-5"></a></span>
<span id="cb35-6"><a href="#cb35-6"></a><span class="kw">library</span>(ggplot2)</span>
<span id="cb35-7"><a href="#cb35-7"></a>caption <-<span class="st"> "Source: Board of Governors of the Federal Reserve System (US)"</span></span>
<span id="cb35-8"><a href="#cb35-8"></a>p <-<span class="st"> </span><span class="kw">ggplot</span>(yield_data, </span>
<span id="cb35-9"><a href="#cb35-9"></a> <span class="kw">aes</span>(<span class="dt">x =</span> DATE, <span class="dt">y =</span> yield_rate, <span class="dt">colour =</span> term)</span>
<span id="cb35-10"><a href="#cb35-10"></a> ) <span class="op">+</span></span>
<span id="cb35-11"><a href="#cb35-11"></a><span class="st"> </span><span class="kw">geom_line</span>() <span class="op">+</span></span>
<span id="cb35-12"><a href="#cb35-12"></a><span class="st"> </span><span class="kw">labs</span>(<span class="dt">x =</span> <span class="st">""</span>, <span class="dt">colour =</span> <span class="st">"Term"</span>, <span class="dt">y =</span> <span class="st">"Percent"</span>,<span class="dt">caption =</span> caption) <span class="op">+</span></span>
<span id="cb35-13"><a href="#cb35-13"></a><span class="st"> </span><span class="kw">ggtitle</span>(<span class="st">"1-Year and 10-Year Treasury Yield Curve Rates"</span>) <span class="op">+</span><span class="st"> </span></span>
<span id="cb35-14"><a href="#cb35-14"></a><span class="st"> </span><span class="kw">theme_bw</span>()</span></code></pre></div>
</section><section id="ggplot2-plot" class="slide level2">
<h2>ggplot2 Plot 📈</h2>
<p><img src="intro-to-r-for-cs1_files/figure-revealjs/ggplot-1.png" width="768" /></p>
</section></section>
<section><section id="cs1-actuarial-examples" class="title-slide slide level1" data-background="intro-to-r-for-cs1_files/assets/galaxy.jpg"><h1>CS1 Actuarial Examples</h1></section><section id="cs1-question-1" class="slide level2 leftAligned">
<h2>CS1 Question 1 📔</h2>
<ul>
<li class="fragment"><em>Given a sample from <span class="math inline">\(X \sim \mathcal{N}(\mu,\,\sigma^{2})\)</span>,</em></li>
</ul>
<ol type="1">
<li class="fragment"><p><em>Calculate the sample mean, <span class="math inline">\(\bar{x}\)</span></em></p></li>
<li class="fragment"><p><em>Calculate a 95% confidence interval (“CI”) for <span class="math inline">\(\mathbb{E}[X]\)</span></em></p></li>
<li class="fragment"><p><em>Calculate a 95% CI for <span class="math inline">\(\mathbb{E}[X]\)</span> using a bootstrap method with a sample size of 10,000</em></p></li>
</ol>
</section><section id="cs1-question-1-answer-part-1" class="slide level2 leftAligned">
<h2>CS1 Question 1: Answer (Part 1) 📖</h2>
<div class="sourceCode" id="cb36"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb36-1"><a href="#cb36-1"></a><span class="co"># We have X equal to:</span></span>
<span id="cb36-2"><a href="#cb36-2"></a>x <-<span class="st"> </span></span>
<span id="cb36-3"><a href="#cb36-3"></a><span class="st"> </span><span class="kw">c</span>(<span class="fl">0.20</span>, <span class="fl">0.40</span>, <span class="fl">1.20</span>, <span class="fl">0.55</span>, <span class="fl">0.75</span>, </span>
<span id="cb36-4"><a href="#cb36-4"></a> <span class="fl">1.10</span>, <span class="fl">0.50</span>, <span class="fl">0.70</span>, <span class="fl">0.55</span>, <span class="fl">0.60</span>, </span>
<span id="cb36-5"><a href="#cb36-5"></a> <span class="fl">0.65</span>, <span class="fl">0.55</span>, <span class="fl">0.50</span>, <span class="fl">0.50</span>, <span class="fl">0.45</span>, </span>
<span id="cb36-6"><a href="#cb36-6"></a> <span class="fl">0.65</span>, <span class="fl">0.55</span>, <span class="fl">0.30</span>, <span class="fl">0.50</span>, <span class="fl">0.45</span></span>
<span id="cb36-7"><a href="#cb36-7"></a> )</span>
<span id="cb36-8"><a href="#cb36-8"></a></span>
<span id="cb36-9"><a href="#cb36-9"></a><span class="co"># 1. Expected value of X is:</span></span>
<span id="cb36-10"><a href="#cb36-10"></a><span class="kw">mean</span>(x)</span></code></pre></div>
<pre><code>## [1] 0.5825</code></pre>
</section><section id="cs1-question-1-answer-part-2" class="slide level2 leftAligned">
<h2>CS1 Question 1: Answer (Part 2) 📖</h2>
<p>Given <span class="math inline">\(X \sim \mathcal{N}(\mu,\,\sigma^{2})\)</span> with unknown population variance, a 95% CI for <span class="math inline">\(\mathbb{E}[X]\)</span> is:</p>
<div class="sourceCode" id="cb38"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb38-1"><a href="#cb38-1"></a>confidence_level <-<span class="st"> </span><span class="fl">0.95</span></span>
<span id="cb38-2"><a href="#cb38-2"></a>confidence_interval <-<span class="st"> </span></span>
<span id="cb38-3"><a href="#cb38-3"></a><span class="st"> </span><span class="kw">c</span>(</span>
<span id="cb38-4"><a href="#cb38-4"></a> (<span class="dv">1</span> <span class="op">-</span><span class="st"> </span>confidence_level)<span class="op">/</span><span class="dv">2</span>,</span>
<span id="cb38-5"><a href="#cb38-5"></a> <span class="dv">1</span> <span class="op">-</span><span class="st"> </span>(<span class="dv">1</span> <span class="op">-</span><span class="st"> </span>confidence_level)<span class="op">/</span><span class="dv">2</span></span>
<span id="cb38-6"><a href="#cb38-6"></a> )</span>
<span id="cb38-7"><a href="#cb38-7"></a>degrees_freedom <-<span class="st"> </span><span class="kw">length</span>(x) <span class="op">-</span><span class="st"> </span><span class="dv">1</span></span>
<span id="cb38-8"><a href="#cb38-8"></a>critical_value <-<span class="st"> </span><span class="kw">qt</span>(confidence_interval, degrees_freedom)</span>
<span id="cb38-9"><a href="#cb38-9"></a>mu <-<span class="st"> </span><span class="kw">mean</span>(x)</span>
<span id="cb38-10"><a href="#cb38-10"></a>sigma <-<span class="st"> </span><span class="kw">sqrt</span>(<span class="kw">var</span>(x))</span>
<span id="cb38-11"><a href="#cb38-11"></a></span>
<span id="cb38-12"><a href="#cb38-12"></a><span class="kw">c</span>(</span>
<span id="cb38-13"><a href="#cb38-13"></a> mu <span class="op">-</span><span class="st"> </span><span class="kw">abs</span>(critical_value[<span class="dv">1</span>]) <span class="op">*</span><span class="st"> </span>sigma <span class="op">/</span><span class="st"> </span><span class="kw">sqrt</span>(<span class="kw">length</span>(x)),</span>
<span id="cb38-14"><a href="#cb38-14"></a> mu <span class="op">+</span><span class="st"> </span><span class="kw">abs</span>(critical_value[<span class="dv">1</span>]) <span class="op">*</span><span class="st"> </span>sigma <span class="op">/</span><span class="st"> </span><span class="kw">sqrt</span>(<span class="kw">length</span>(x))</span>
<span id="cb38-15"><a href="#cb38-15"></a>)</span></code></pre></div>
<pre><code>## [1] 0.4737234 0.6912766</code></pre>
</section><section id="cs1-question-1-answer-part-3" class="slide level2 leftAligned">
<h2>CS1 Question 1: Answer (Part 3) 📖</h2>
<p>We can also calculate a 95% CI for <span class="math inline">\(\mathbb{E}[X]\)</span> using a bootstrap method:</p>
<div class="sourceCode" id="cb40"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb40-1"><a href="#cb40-1"></a><span class="kw">set.seed</span>(<span class="dv">42</span>)</span>
<span id="cb40-2"><a href="#cb40-2"></a>sample_size <-<span class="st"> </span><span class="dv">10000</span></span>
<span id="cb40-3"><a href="#cb40-3"></a>confidence_level <-<span class="st"> </span><span class="fl">0.95</span></span>
<span id="cb40-4"><a href="#cb40-4"></a>confidence_interval <-<span class="st"> </span></span>
<span id="cb40-5"><a href="#cb40-5"></a><span class="st"> </span><span class="kw">c</span>(</span>
<span id="cb40-6"><a href="#cb40-6"></a> (<span class="dv">1</span> <span class="op">-</span><span class="st"> </span>confidence_level)<span class="op">/</span><span class="dv">2</span>,</span>
<span id="cb40-7"><a href="#cb40-7"></a> <span class="dv">1</span> <span class="op">-</span><span class="st"> </span>(<span class="dv">1</span> <span class="op">-</span><span class="st"> </span>confidence_level)<span class="op">/</span><span class="dv">2</span></span>
<span id="cb40-8"><a href="#cb40-8"></a> )</span>
<span id="cb40-9"><a href="#cb40-9"></a>x_bootstrapped <-<span class="st"> </span></span>
<span id="cb40-10"><a href="#cb40-10"></a><span class="st"> </span><span class="kw">replicate</span>(sample_size,</span>
<span id="cb40-11"><a href="#cb40-11"></a> <span class="kw">mean</span>(<span class="kw">sample</span>(x, <span class="dt">replace =</span> <span class="ot">TRUE</span>))</span>
<span id="cb40-12"><a href="#cb40-12"></a> )</span>
<span id="cb40-13"><a href="#cb40-13"></a><span class="kw">quantile</span>(x_bootstrapped, <span class="dt">probs =</span> confidence_interval)</span></code></pre></div>
<pre><code>## 2.5% 97.5%
## 0.49 0.69</code></pre>
</section><section id="cs1-question-2" class="slide level2 leftAligned">
<h2>CS1 Question 2 📔</h2>
<ul>
<li class="fragment"><em>Given the dataset “house_data.csv” (given below)</em>:</li>
</ul>
<ol type="1">
<li class="fragment"><p><em>Fit a generalised linear model to predict house prices using a Gamma distribution</em></p></li>
<li class="fragment"><p><em>State the formula of the model fitted</em></p></li>
<li class="fragment"><p><em>Review the model fit</em></p></li>
</ol>
<div class="sourceCode" id="cb42"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb42-1"><a href="#cb42-1"></a>house_data <-<span class="st"> </span><span class="kw">read.csv</span>(<span class="st">"intro-to-r-for-cs1_files/data/house_data.csv"</span>)</span>
<span id="cb42-2"><a href="#cb42-2"></a><span class="kw">head</span>(house_data[,<span class="op">-</span><span class="kw">c</span>(<span class="dv">1</span>,<span class="dv">8</span><span class="op">:</span><span class="dv">11</span>)]) <span class="co"># Glimpse of data set</span></span></code></pre></div>
<pre><code>## Region YearBuilt RoofType HouseSize Bathrooms Bedrooms
## 1 5 2003 1 1710 3.5 3
## 2 5 2002 1 1786 3.5 3
## 3 1 1965 2 1040 2.0 3
## 4 1 1962 2 912 2.0 2
## 5 6 1960 2 1253 2.5 2
## 6 6 1970 1 1004 2.0 2</code></pre>
</section><section id="cs1-question-2-answer-part-1" class="slide level2 leftAligned">
<h2>CS1 Question 2: Answer (Part 1) 📖</h2>
<p>Let’s build a GLM with response variables “<em>YearBuilt</em>”, “<em>HouseSize</em>” and “<em>RoofType</em>”:</p>
<div class="sourceCode" id="cb44"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1"></a>glm_model <-<span class="st"> </span><span class="kw">glm</span>(</span>
<span id="cb44-2"><a href="#cb44-2"></a> Price <span class="op">~</span><span class="st"> </span>YearBuilt <span class="op">+</span><span class="st"> </span>HouseSize <span class="op">+</span><span class="st"> </span>RoofType,</span>
<span id="cb44-3"><a href="#cb44-3"></a> <span class="dt">family =</span> <span class="kw">Gamma</span>(<span class="dt">link =</span> <span class="st">"inverse"</span>),</span>
<span id="cb44-4"><a href="#cb44-4"></a> <span class="dt">data =</span> house_data</span>
<span id="cb44-5"><a href="#cb44-5"></a>)</span></code></pre></div>
</section><section id="cs1-question-2-answer-part-2" class="slide level2 leftAligned">
<h2>CS1 Question 2: Answer (Part 2) 📖</h2>
<div class="sourceCode" id="cb45"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb45-1"><a href="#cb45-1"></a><span class="kw">coef</span>(glm_model)</span></code></pre></div>
<pre><code>## (Intercept) YearBuilt HouseSize RoofType
## 7.829688e-05 -3.492117e-08 -2.006639e-09 6.302872e-09</code></pre>
<p>We have fit a model with <span class="math inline">\(\bar{\nu} = \frac{1}{\mu} =\)</span> 12,772 <span class="math inline">\(-\)</span> 28,635,925<span class="math inline">\(x_1 \,-\)</span> 498,345,682<span class="math inline">\(x_2 \,+\)</span> 158,657,830<span class="math inline">\(x_3\)</span></p>
<p>where <span class="math inline">\(\bar{\nu}\)</span> is the predicted variable <em>“Price”</em> and <span class="math inline">\(x_1,\, x_2,\, x_3\)</span> are the <em>“YearBuilt”</em>, <em>“HouseSize”</em> and <em>“RoofType”</em> variables respectively.</p>
</section><section id="cs1-question-2-answer-part-3" class="slide level2 leftAligned">
<h2>CS1 Question 2: Answer (Part 3) 📖</h2>
<div class="sourceCode" id="cb47"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb47-1"><a href="#cb47-1"></a><span class="kw">summary</span>(glm_model)</span></code></pre></div>
<pre><code>##
## Call:
## glm(formula = Price ~ YearBuilt + HouseSize + RoofType, family = Gamma(link = "inverse"),
## data = house_data)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.74826 -0.09318 0.01491 0.10941 0.66029
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.830e-05 5.274e-06 14.845 <2e-16 ***
## YearBuilt -3.492e-08 2.667e-09 -13.094 <2e-16 ***
## HouseSize -2.007e-09 8.738e-11 -22.965 <2e-16 ***
## RoofType 6.303e-09 8.191e-09 0.769 0.442
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1