diff --git a/docs/articles/Tutorials.html b/docs/articles/Tutorials.html
index e54b419..46c3929 100644
--- a/docs/articles/Tutorials.html
+++ b/docs/articles/Tutorials.html
@@ -83,7 +83,7 @@ Tutorials for k-means clustering inference
-We first generate data according to \(\mathbf{X} \sim {MN}_{n\times q}(\boldsymbol{\mu},
+We first generate data according to \(\mathbf{X} \sim MN_{n\times q}(\boldsymbol{\mu},
\textbf{I}_n, \sigma^2 \textbf{I}_q)\) with \(n=150,q=2,\sigma=1,\) and \[\begin{align}
\label{eq:power_model}
\boldsymbol{\mu}_1 =\ldots = \boldsymbol{\mu}_{50} = \begin{bmatrix}
@@ -95,9 +95,9 @@ Tutorials for k-means clustering inference
\boldsymbol{\mu}_{101}=\ldots = \boldsymbol{\mu}_{150} = \begin{bmatrix}
\delta/2 \\ 0_{q-1}
\end{bmatrix}.
-\end{align}\] Here, we can think of \({C}_1 = \{1,\ldots,50\},{C}_2 =
-\{51,\ldots,100\},{C}_3 = \{101,\ldots,150\}\) as the “true
-clusters”. In the figure below, we display one such simulation \(\mathbf{x}\in\mathbb{R}^{100\times 2}\)
+\end{align}\]
Here, we can think of \(C_1 = \{1,\ldots,50\},C_2 = \{51,\ldots,100\},C_3
+= \{101,\ldots,150\}\) as the “true clusters”. In the figure
+below, we display one such simulation \(\mathbf{x}\in\mathbb{R}^{100\times 2}\)
with \(\delta=10\).
set.seed(2022)
@@ -187,9 +187,8 @@ Inference
#> cluster_1 cluster_2 test_stat p_selective p_naive
#> 1 2 3 4.464756 8.514513e-29 2.171388e-110
In the summary, we have the empirical difference in means of the
-second feature between the two clusters, i.e.,\(\sum_{i\in
-{\hat{{G}}}}\mathbf{x}_{i,2}/|\hat{{G}}| - \sum_{i\in
-\hat{G}'}\mathbf{x}_{i,2}/|\hat{G}'|\)
+second feature between the two clusters, i.e.,\(\sum_{i\in \hat{G}}\mathbf{x}_{i,2}/|\hat{{G}}| -
+\sum_{i\in \hat{G}'}\mathbf{x}_{i,2}/|\hat{G}'|\)
(test_stats), the naive p-value based on a z-test
(p_naive), and the selective \(p\)-value (p_selective). In
this case, the test based on \(p_{\text{selective}}\) can reject this null
@@ -221,13 +220,12 @@
Inferen
#> 2 0 0 50 0
#> 3 25 0 0 25
By inspection, we see that the blue clusters (labeled as cluster 1)
-and the grey clusters (labeled as cluster 4) have the same mean. Now
-\(p_{\text{selective}}\) yields a much
+and the grey clusters (labeled as cluster 4) have the same mean. Now the
+selective \(p\)-value yields a much
more moderate \(p\)-value, and the test
-based on \(p_{2,\text{selective}}\)
-cannot reject the null hypothesis when it holds. By contrast, the naive
-\(p\)-value is tiny and leads to an
-anti-conservative test.
+based on it cannot reject the null hypothesis when it holds. By
+contrast, the naive \(p\)-value is tiny
+and leads to an anti-conservative test.
cluster_1 <- 1
cluster_2 <- 4
diff --git a/docs/articles/Tutorials_hier.html b/docs/articles/Tutorials_hier.html
index 59677d3..4ac46b2 100644
--- a/docs/articles/Tutorials_hier.html
+++ b/docs/articles/Tutorials_hier.html
@@ -156,10 +156,10 @@ Infer
cluster_1 <- 1
cluster_2 <- 3
-cl_1_2_inference_demo <- test_hier_clusters_exact_1f(X=X, link="average", hcl=hcl, K=3, k1=1, k2=2, feat=1)
-summary(cl_1_2_inference_demo)
+cl_inference_demo <- test_hier_clusters_exact_1f(X=X, link="average", hcl=hcl, K=3, k1=cluster_1, k2=cluster_2, feat=1)
+summary(cl_inference_demo)
#> cluster_1 cluster_2 test_stat p_selective p_naive
-#> 1 1 2 4.464756 8.870985e-08 1.774197e-07
In the summary, we have the empirical difference in means of the
first feature between the two clusters, i.e.,\(\sum_{i\in
{\hat{{G}}}}\mathbf{x}_{i,2}/|\hat{{G}}| - \sum_{i\in
@@ -179,15 +179,17 @@ \(p_{2,\text{selective}}\)
-yields a much more moderate \(p\)-value, and the test based on \(p_{2,\text{selective}}\) cannot reject the
-null hypothesis when it holds. By contrast, the naive \(p\)-value is tiny and leads to an
-anti-conservative test.
+Now the selective \(p\)-value yields
+a much more moderate \(p\)-value, and
+the test based on it cannot reject the null hypothesis when it holds. By
+contrast, the naive \(p\)-value is tiny
+and leads to an anti-conservative test.
cluster_1 <- 1
-cluster_2 <- 4
-cl_1_2_inference_demo <- test_hier_clusters_exact_1f(X=X, link="average", hcl=hcl, K=3, k1=1, k2=3, feat=2)
-summary(cl_1_2_inference_demo)
+cluster_2 <- 3
+cl_inference_demo <- test_hier_clusters_exact_1f(X=X, link="average",
+ hcl=hcl, K=3, k1=cluster_1, k2=cluster_2, feat=2)
+summary(cl_inference_demo)
#> cluster_1 cluster_2 test_stat p_selective p_naive
#> 1 1 3 -0.1766818 0.8362984 0.8362984
Technical details
-Figure 1: We simulated one dataset according to \({MN}_{100\times 10}(\mu, \textbf{I}_{100},
+Figure 1: We simulated one dataset according to \(MN_{100\times 10}(\mu, \textbf{I}_{100},
\Sigma)\), where \(\mu_i =
(1,0_9)^T\) for \(i=1,\ldots,
50\) and \(\mu_i = (0_9,1)^T\)
@@ -160,8 +160,8 @@ \(p_{j, \text{selective}}\)
, we have
-conditioned on (i) the estimated clusters \(\mathcal{C}(x)\) to account for the
-data-driven nature of the null hypothesis; and (ii) \(U(x)\) to eliminate the unknown nuisance
+conditioned on (i) the estimated clusters \({C}(x)\) to account for the data-driven
+nature of the null hypothesis; and (ii) \(U(x)\) to eliminate the unknown nuisance
parameters under the null.
We show that this \(p\)-value for
testing \(\hat{H}_{0j}\) can be written
@@ -169,21 +169,22 @@
\(\mathbb{F}(t;
-\mu, \sigma, \mathcal{S})\) denotes the cumulative distribution
-function (CDF) of a \(N(\mu,
-\sigma^2)\) random variable truncated to the set \(\mathcal{S}\), \(x'(\phi,j) = x + (\phi - ( \bar{x}_{\hat{G}j}
-- \bar{x}_{\hat{G'}j})) \left ( \frac{ \hat{\nu} }{
-\|\hat{\nu}\|_2^2 } \right ) \left ( \frac{\Sigma_j}{\Sigma_{jj}} \right
-)^T,\) and {\[\begin{equation}
+\mu, \sigma, {S})\) denotes the cumulative distribution function
+(CDF) of a \(N(\mu, \sigma^2)\) random
+variable truncated to the set \({S}\),
+\(x'(\phi,j) = x + (\phi - (
+\bar{x}_{\hat{G}j} - \bar{x}_{\hat{G'}j})) \left ( \frac{ \hat{\nu}
+}{ \|\hat{\nu}\|_2^2 } \right ) \left ( \frac{\Sigma_j}{\Sigma_{jj}}
+\right )^T,\) and \[\begin{equation}
\hat S_j =
-\left \{ \phi \in \mathbb{R}: C(x) =
-\mathcal{C}\left(x'(\phi,j)\right ) \right \}. \label{eq:defS}
-\end{equation}\] }.
+\left \{ \phi \in \mathbb{R}: C(x) = {C}\left(x'(\phi,j)\right )
+\right \}. \label{eq:defS}
+\end{equation}\].
While the notation in the last paragraph might seem daunting, the
intuition is simple: since \(p_{j,
\text{selective}}\) can be rewritten into sums of CDFs of
diff --git a/docs/pkgdown.yml b/docs/pkgdown.yml
index 7ba3443..dc3b4ec 100644
--- a/docs/pkgdown.yml
+++ b/docs/pkgdown.yml
@@ -5,7 +5,7 @@ articles:
Tutorials: Tutorials.html
Tutorials_hier: Tutorials_hier.html
technical_details: technical_details.html
-last_built: 2023-11-26T01:22Z
+last_built: 2023-11-26T01:39Z
urls:
reference: https://yiqunchen.github.io/CADET/reference
article: https://yiqunchen.github.io/CADET/articles
diff --git a/vignettes/Tutorials.Rmd b/vignettes/Tutorials.Rmd
index 273aaca..52b9c6c 100644
--- a/vignettes/Tutorials.Rmd
+++ b/vignettes/Tutorials.Rmd
@@ -23,7 +23,7 @@ library(CADET)
library(ggplot2)
```
-We first generate data according to $\mathbf{X} \sim {MN}_{n\times q}(\boldsymbol{\mu}, \textbf{I}_n, \sigma^2 \textbf{I}_q)$ with $n=150,q=2,\sigma=1,$ and
+We first generate data according to $\mathbf{X} \sim MN_{n\times q}(\boldsymbol{\mu}, \textbf{I}_n, \sigma^2 \textbf{I}_q)$ with $n=150,q=2,\sigma=1,$ and
\begin{align}
\label{eq:power_model}
\boldsymbol{\mu}_1 =\ldots = \boldsymbol{\mu}_{50} = \begin{bmatrix}
@@ -35,7 +35,7 @@ We first generate data according to $\mathbf{X} \sim {MN}_{n\times q}(\boldsymbo
\delta/2 \\ 0_{q-1}
\end{bmatrix}.
\end{align}
-Here, we can think of ${C}_1 = \{1,\ldots,50\},{C}_2 = \{51,\ldots,100\},{C}_3 = \{101,\ldots,150\}$ as the "true clusters".
+Here, we can think of $C_1 = \{1,\ldots,50\},C_2 = \{51,\ldots,100\},C_3 = \{101,\ldots,150\}$ as the "true clusters".
In the figure below, we display one such simulation $\mathbf{x}\in\mathbb{R}^{100\times 2}$ with $\delta=10$.
```{r fig.align="center", fig.height = 5, fig.width = 5}
@@ -98,7 +98,7 @@ cl_inference_demo <- kmeans_inference_1f(X, k=3, cluster_1, cluster_2,
summary(cl_inference_demo)
```
-In the summary, we have the empirical difference in means of the second feature between the two clusters, i.e.,$\sum_{i\in {\hat{{G}}}}\mathbf{x}_{i,2}/|\hat{{G}}| - \sum_{i\in \hat{G}'}\mathbf{x}_{i,2}/|\hat{G}'|$ (`test_stats`), the naive p-value based on a z-test (`p_naive`), and the selective $p$-value (`p_selective`). In this case, the test based on $p_{\text{selective}}$ can reject this null hypothesis that the blue and pink clusters have the same mean in the first feature ($p_{2,\text{selective}}<0.001$).
+In the summary, we have the empirical difference in means of the second feature between the two clusters, i.e.,$\sum_{i\in \hat{G}}\mathbf{x}_{i,2}/|\hat{{G}}| - \sum_{i\in \hat{G}'}\mathbf{x}_{i,2}/|\hat{G}'|$ (`test_stats`), the naive p-value based on a z-test (`p_naive`), and the selective $p$-value (`p_selective`). In this case, the test based on $p_{\text{selective}}$ can reject this null hypothesis that the blue and pink clusters have the same mean in the first feature ($p_{2,\text{selective}}<0.001$).
### Inference for k-means clustering when the null hypothesis holds
diff --git a/vignettes/technical_details.Rmd b/vignettes/technical_details.Rmd
index c73ae4d..cf1a0dd 100644
--- a/vignettes/technical_details.Rmd
+++ b/vignettes/technical_details.Rmd
@@ -19,7 +19,7 @@ knitr::opts_chunk$set(
{width=90%}
-Figure 1: We simulated one dataset according to ${MN}_{100\times 10}(\mu, \textbf{I}_{100}, \Sigma)$, where $\mu_i = (1,0_9)^T$ for $i=1,\ldots, 50$ and $\mu_i = (0_9,1)^T$ for $i=51,\ldots, 100$, and $\Sigma_{ij} = 1\{i=j\}+0.4\cdot 1\{i\neq j\}$. *(a)*: Empirical distribution of feature 2 based on the simulated data set. In this case, all observations have the same mean for feature 2. *(b)*: We apply k-means clustering to obtain two clusters and plot the empirical distribution of feature 2 stratified by the clusters. *(c)*: Quantile-quantile plot of naive z-test (black) our proposed p-values (orange) applied to the simulated data sets for testing the null hypotheses for a difference in means for features 2--8.
+Figure 1: We simulated one dataset according to $MN_{100\times 10}(\mu, \textbf{I}_{100}, \Sigma)$, where $\mu_i = (1,0_9)^T$ for $i=1,\ldots, 50$ and $\mu_i = (0_9,1)^T$ for $i=51,\ldots, 100$, and $\Sigma_{ij} = 1\{i=j\}+0.4\cdot 1\{i\neq j\}$. *(a)*: Empirical distribution of feature 2 based on the simulated data set. In this case, all observations have the same mean for feature 2. *(b)*: We apply k-means clustering to obtain two clusters and plot the empirical distribution of feature 2 stratified by the clusters. *(c)*: Quantile-quantile plot of naive z-test (black) our proposed p-values (orange) applied to the simulated data sets for testing the null hypotheses for a difference in means for features 2--8.