<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta charset="utf-8" />
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="pandoc" />
<meta name="author" content="Yuxin Zou" />
<title>Estimate cor — MLE EM rho</title>
<script src="site_libs/jquery-1.11.3/jquery.min.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link href="site_libs/bootstrap-3.3.5/css/cosmo.min.css" rel="stylesheet" />
<script src="site_libs/bootstrap-3.3.5/js/bootstrap.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/html5shiv.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/respond.min.js"></script>
<script src="site_libs/jqueryui-1.11.4/jquery-ui.min.js"></script>
<link href="site_libs/tocify-1.9.1/jquery.tocify.css" rel="stylesheet" />
<script src="site_libs/tocify-1.9.1/jquery.tocify.js"></script>
<script src="site_libs/navigation-1.1/tabsets.js"></script>
<script src="site_libs/navigation-1.1/codefolding.js"></script>
<link href="site_libs/highlightjs-9.12.0/textmate.css" rel="stylesheet" />
<script src="site_libs/highlightjs-9.12.0/highlight.js"></script>
<link href="site_libs/font-awesome-5.0.13/css/fa-svg-with-js.css" rel="stylesheet" />
<script src="site_libs/font-awesome-5.0.13/js/fontawesome-all.min.js"></script>
<script src="site_libs/font-awesome-5.0.13/js/fa-v4-shims.min.js"></script>
<style type="text/css">code{white-space: pre;}</style>
<style type="text/css">
pre:not([class]) {
background-color: white;
}
</style>
<script type="text/javascript">
if (window.hljs) {
hljs.configure({languages: []});
hljs.initHighlightingOnLoad();
if (document.readyState && document.readyState === "complete") {
window.setTimeout(function() { hljs.initHighlighting(); }, 0);
}
}
</script>
<style type="text/css">
h1 {
font-size: 34px;
}
h1.title {
font-size: 38px;
}
h2 {
font-size: 30px;
}
h3 {
font-size: 24px;
}
h4 {
font-size: 18px;
}
h5 {
font-size: 16px;
}
h6 {
font-size: 12px;
}
.table th:not([align]) {
text-align: left;
}
</style>
</head>
<body>
<style type = "text/css">
.main-container {
max-width: 940px;
margin-left: auto;
margin-right: auto;
}
code {
color: inherit;
background-color: rgba(0, 0, 0, 0.04);
}
img {
max-width:100%;
height: auto;
}
.tabbed-pane {
padding-top: 12px;
}
.html-widget {
margin-bottom: 20px;
}
button.code-folding-btn:focus {
outline: none;
}
</style>
<style type="text/css">
/* padding for bootstrap navbar */
body {
padding-top: 51px;
padding-bottom: 40px;
}
/* offset scroll position for anchor links (for fixed navbar) */
.section h1 {
padding-top: 56px;
margin-top: -56px;
}
.section h2 {
padding-top: 56px;
margin-top: -56px;
}
.section h3 {
padding-top: 56px;
margin-top: -56px;
}
.section h4 {
padding-top: 56px;
margin-top: -56px;
}
.section h5 {
padding-top: 56px;
margin-top: -56px;
}
.section h6 {
padding-top: 56px;
margin-top: -56px;
}
</style>
<script>
// manage active state of menu based on current page
$(document).ready(function () {
// active menu anchor
href = window.location.pathname
href = href.substr(href.lastIndexOf('/') + 1)
if (href === "")
href = "index.html";
var menuAnchor = $('a[href="' + href + '"]');
// mark it active
menuAnchor.parent().addClass('active');
// if it's got a parent navbar menu mark it active as well
menuAnchor.closest('li.dropdown').addClass('active');
});
</script>
<div class="container-fluid main-container">
<!-- tabsets -->
<script>
$(document).ready(function () {
window.buildTabsets("TOC");
});
</script>
<!-- code folding -->
<style type="text/css">
.code-folding-btn { margin-bottom: 4px; }
</style>
<script>
$(document).ready(function () {
window.initializeCodeFolding("hide" === "show");
});
</script>
<script>
$(document).ready(function () {
// move toc-ignore selectors from section div to header
$('div.section.toc-ignore')
.removeClass('toc-ignore')
.children('h1,h2,h3,h4,h5').addClass('toc-ignore');
// establish options
var options = {
selectors: "h1,h2,h3",
theme: "bootstrap3",
context: '.toc-content',
hashGenerator: function (text) {
return text.replace(/[.\\/?&!#<>]/g, '').replace(/\s/g, '_').toLowerCase();
},
ignoreSelector: ".toc-ignore",
scrollTo: 0
};
options.showAndHide = true;
options.smoothScroll = true;
// tocify
var toc = $("#TOC").tocify(options).data("toc-tocify");
});
</script>
<style type="text/css">
#TOC {
margin: 25px 0px 20px 0px;
}
@media (max-width: 768px) {
#TOC {
position: relative;
width: 100%;
}
}
.toc-content {
padding-left: 30px;
padding-right: 40px;
}
div.main-container {
max-width: 1200px;
}
div.tocify {
width: 20%;
max-width: 260px;
max-height: 85%;
}
@media (min-width: 768px) and (max-width: 991px) {
div.tocify {
width: 25%;
}
}
@media (max-width: 767px) {
div.tocify {
width: 100%;
max-width: none;
}
}
.tocify ul, .tocify li {
line-height: 20px;
}
.tocify-subheader .tocify-item {
font-size: 0.90em;
padding-left: 25px;
text-indent: 0;
}
.tocify .list-group-item {
border-radius: 0px;
}
</style>
<!-- setup 3col/9col grid for toc_float and main content -->
<div class="row-fluid">
<div class="col-xs-12 col-sm-4 col-md-3">
<div id="TOC" class="tocify">
</div>
</div>
<div class="toc-content col-xs-12 col-sm-8 col-md-9">
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">mash_application</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="index.html">Home</a>
</li>
<li>
<a href="about.html">About</a>
</li>
<li>
<a href="license.html">License</a>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li>
<a href="https://github.com/jdblischak/workflowr">
<span class="fa fa-github"></span>
</a>
</li>
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container -->
</div><!--/.navbar -->
<!-- Add a small amount of space between sections. -->
<style type="text/css">
div.section {
padding-top: 12px;
}
</style>
<div class="fluid-row" id="header">
<div class="btn-group pull-right">
<button type="button" class="btn btn-default btn-xs dropdown-toggle" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"><span>Code</span> <span class="caret"></span></button>
<ul class="dropdown-menu" style="min-width: 50px;">
<li><a id="rmd-show-all-code" href="#">Show All Code</a></li>
<li><a id="rmd-hide-all-code" href="#">Hide All Code</a></li>
</ul>
</div>
<h1 class="title toc-ignore">Estimate cor — MLE EM rho</h1>
<h4 class="author"><em>Yuxin Zou</em></h4>
<h4 class="date"><em>2018-8-3</em></h4>
</div>
<!-- Update knitr chunk options -->
<!-- Insert the date the file was last updated -->
<p><strong>Last updated:</strong> 2018-08-15</p>
<pre class="r"><code>library(mashr)</code></pre>
<pre><code>Loading required package: ashr</code></pre>
<pre class="r"><code>library(knitr)
library(kableExtra)
source('../code/estimate_cor.R')
source('../code/generateDataV.R')
source('../code/summary.R')</code></pre>
<div id="em" class="section level1">
<h1>EM</h1>
<p>We use EM algorithm to update <span class="math inline">\(\rho\)</span>.</p>
<div id="e-step" class="section level2">
<h2>E step</h2>
<p><span class="math display">\[
P(\hat{B},B,Z|\rho, \pi) = \prod_{i=1}^{n} \prod_{k=0}^{K}\left[\pi_{k}N(\hat{b}_{i}; b_{i}, V)N(b_{i}; 0, U_{k})\right]^{\mathbb{I}(z_{i}=k)}
\]</span></p>
<p><span class="math display">\[
\mathbb{E}_{Z,B|\hat{B}} \log P(\hat{B},B,Z|\rho, \pi) = \sum_{i=1}^{n} \sum_{k=0}^{K} P(z_{i}=k|\hat{b}_{i})\left[ \log \pi_{k} + \mathbb{E}_{B|\hat{B}}(\log N(\hat{b}_{i}; b_{i}, V)) + \mathbb{E}_{B|\hat{B}}(\log N(b_{i}; 0, U_{k})) \right]
\]</span></p>
<p><span class="math display">\[
\begin{align*}
\log N(\hat{b}_{i}; b_{i}, V) + \log N(b_{i}; 0, U_{k}) &= -\frac{p}{2}\log 2\pi -\frac{1}{2}\log |V| - \frac{1}{2}(\hat{b}_{i}-b_{i})^{T}V^{-1}(\hat{b}_{i}-b_{i}) -\frac{p}{2}\log 2\pi -\frac{1}{2}\log |U_{k}| - \frac{1}{2}b_{i}^{T}U_{k}^{-1}b_{i} \\
&= -p\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log |V| - \frac{1}{2}\hat{b}_{i}^{T}V^{-1}\hat{b}_{i} + \hat{b}_{i}^{T}V^{-1}b_{i} -\frac{1}{2}b_{i}^{T}V^{-1}b_{i} - \frac{1}{2}b_{i}^{T}U_{k}^{-1}b_{i} \\
\mathbb{E}_{b_{i}|\hat{b}_{i}}\left[ \log N(\hat{b}_{i}; b_{i}, V) + \log N(b_{i}; 0, U_{k}) \right] &= -p\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log |V| - \frac{1}{2}\hat{b}_{i}^{T}V^{-1}\hat{b}_{i} + \hat{b}_{i}^{T}V^{-1}\mathbb{E}(b_{i}|\hat{b}_{i}) -\frac{1}{2}tr\left(V^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) - \frac{1}{2}tr\left(U_{k}^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right)
\end{align*}
\]</span> <span class="math display">\[
V = \left( \begin{matrix}1 & \rho \\ \rho & 1 \end{matrix} \right)
\]</span> Let <span class="math inline">\(\mu_{i} = \mathbb{E}(b_{i}|\hat{b}_{i})\)</span> <span class="math display">\[
\begin{align*}
\mathbb{E}_{b_{i}|\hat{b}_{i}}\left[ \log N(\hat{b}_{i}; b_{i}, V) + \log N(b_{i}; 0, U_{k}) \right] &= -2\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log(1-\rho^2) - \frac{1}{2(1-\rho^2)}\left(\hat{b}_{i1}^2 + \hat{b}_{i2}^2 -2\hat{b}_{i1} \mu_{i1} -2\hat{b}_{i2} \mu_{i2} + \mathbb{E}(b_{i1}^2|\hat{b}_{i}) + \mathbb{E}(b_{i2}^2|\hat{b}_{i}) - 2\hat{b}_{i1}\hat{b}_{i2}\rho + 2 \hat{b}_{i1}\mu_{i2}\rho +2\hat{b}_{i2}\mu_{i1}\rho - 2\rho\mathbb{E}(b_{i1}b_{i2}|\hat{b}_{i}) \right) - \frac{1}{2}tr\left(U_{k}^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right)
\end{align*}
\]</span></p>
<p><span class="math display">\[
\gamma_{Z_{i}}(k) = P(z_{i}=k|X_{i}) = \frac{\pi_{k}N(x_{i}; 0, V+U_{k})}{\sum_{k'=0}^{K}\pi_{k'}N(x_{i}; 0, V + U_{k'})}
\]</span></p>
</div>
<div id="m-step" class="section level2">
<h2>M step</h2>
<p><span class="math inline">\(V\)</span>: <span class="math display">\[
\begin{align*}
f(V^{-1}) = \sum_{i=1}^{n} \sum_{k=0}^{K} \gamma_{Z_{i}}(k)\left[ -p\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log |V| - \frac{1}{2}\hat{b}_{i}^{T}V^{-1}\hat{b}_{i} + \hat{b}_{i}^{T}V^{-1}\mathbb{E}(b_{i}|\hat{b}_{i}) -\frac{1}{2}tr\left(V^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) - \frac{1}{2}tr\left(U_{k}^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) \right]
\end{align*}
\]</span></p>
<p><span class="math display">\[
\begin{align*}
f(V^{-1})' &= \sum_{i=1}^{n} \sum_{k=0}^{K} \gamma_{Z_{i}}(k)\left[ \frac{1}{2}V - \frac{1}{2}\hat{b}_{i}\hat{b}_{i}^{T} + \mathbb{E}(b_{i}|\hat{b}_{i})\hat{b}_{i}^{T} - \frac{1}{2} \mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right] = 0 \\
\frac{1}{2}Vn &= \sum_{i=1}^{n} \sum_{k=0}^{K} \gamma_{Z_{i}}(k)\left[\frac{1}{2}\hat{b}_{i}\hat{b}_{i}^{T} - \mathbb{E}(b_{i}|\hat{b}_{i})\hat{b}_{i}^{T} + \frac{1}{2} \mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right] \\
\hat{V} &= \frac{1}{n} \sum_{i=1}^{n} \left[\hat{b}_{i}\hat{b}_{i}^{T} - 2\mathbb{E}(b_{i}|\hat{b}_{i})\hat{b}_{i}^{T} + \mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right]
\end{align*}
\]</span></p>
<p><span class="math inline">\(\rho\)</span>: <span class="math display">\[
f(\rho) = \sum_{i=1}^{n} \sum_{k=1}^{K} \gamma_{Z_{i}}(k)\left[-2\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log(1-\rho^2) - \frac{1}{2(1-\rho^2)}\left(\hat{b}_{i1}^2 + \hat{b}_{i2}^2 -2\hat{b}_{i1} \mu_{i1} -2\hat{b}_{i2} \mu_{i2} + \mathbb{E}(b_{i1}^2|\hat{b}_{i}) + \mathbb{E}(b_{i2}^2|\hat{b}_{i}) - 2\hat{b}_{i1}\hat{b}_{i2}\rho + 2 \hat{b}_{i1}\mu_{i2}\rho +2\hat{b}_{i2}\mu_{i1}\rho - 2\rho\mathbb{E}(b_{i1}b_{i2}|\hat{b}_{i}) \right) - \frac{1}{2}tr\left(U_{k}^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right)\right]
\]</span></p>
<p><span class="math display">\[
\begin{align*}
f(\rho)' = \sum_{i=1}^{n} \sum_{k=1}^{K} \gamma_{Z_{i}}(k)\left[ \frac{\rho}{1-\rho^2} -\frac{\rho}{(1-\rho^2)^2}\left( \hat{b}_{i1}^2 + \hat{b}_{i2}^2 -2\hat{b}_{i1} \mu_{i1} -2\hat{b}_{i2} \mu_{i2} + \mathbb{E}(b_{i1}^2|\hat{b}_{i}) + \mathbb{E}(b_{i2}^2|\hat{b}_{i}) \right) -\frac{\rho^2+1}{(1-\rho^2)^2}\left( -\hat{b}_{i1}\hat{b}_{i2} + \hat{b}_{i1}\mu_{i2} +\hat{b}_{i2}\mu_{i1} - \mathbb{E}(b_{i1}b_{i2}|\hat{b}_{i}) \right) \right] &= 0 \\
\rho(1-\rho^2)n - \rho \sum_{i=1}^{n} \sum_{k=1}^{K} \gamma_{Z_{i}}(k) \left( \hat{b}_{i1}^2 + \hat{b}_{i2}^2 -2\hat{b}_{i1} \mu_{i1} -2\hat{b}_{i2} \mu_{i2} + \mathbb{E}(b_{i1}^2|\hat{b}_{i}) + \mathbb{E}(b_{i2}^2|\hat{b}_{i}) \right) - (\rho^2 + 1) \sum_{i=1}^{n} \sum_{k=1}^{K} \gamma_{Z_{i}}(k)\left( -\hat{b}_{i1}\hat{b}_{i2} + \hat{b}_{i1}\mu_{i2} +\hat{b}_{i2}\mu_{i1} - \mathbb{E}(b_{i1}b_{i2}|\hat{b}_{i}) \right) &= 0 \\
-n\rho^{3} - \rho^2 \sum_{i=1}^{n} \left( -\hat{b}_{i1}\hat{b}_{i2} + \hat{b}_{i1}\mu_{i2} +\hat{b}_{i2}\mu_{i1} - \mathbb{E}(b_{i1}b_{i2}|\hat{b}_{i}) \right) - \rho(\sum_{i=1}^{n} \left( \hat{b}_{i1}^2 + \hat{b}_{i2}^2 -2\hat{b}_{i1} \mu_{i1} -2\hat{b}_{i2} \mu_{i2} + \mathbb{E}(b_{i1}^2|\hat{b}_{i}) + \mathbb{E}(b_{i2}^2|\hat{b}_{i}) \right) - n) - \sum_{i=1}^{n} \left( -\hat{b}_{i1}\hat{b}_{i2} + \hat{b}_{i1}\mu_{i2} +\hat{b}_{i2}\mu_{i1} - \mathbb{E}(b_{i1}b_{i2}|\hat{b}_{i}) \right) &= 0
\end{align*}
\]</span></p>
<p>The polynomial has either 1 or 3 real roots in (-1, 1).</p>
<p>Algorithm:</p>
<pre class="text"><code>Input: X, Ulist, init_rho
Compute loglikelihood
delta = 1
while delta > tol
Given rho, Estimate pi using convex method (current mash method)
M step: update rho: find all roots of polynomial, if it has three real roots, choose the one with higher loglikelihood.
Compute loglikelihood
Update delta</code></pre>
</div>
</div>
<div id="data" class="section level1">
<h1>Data</h1>
<p><span class="math display">\[
\hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\
0.5 & 1 \end{matrix}\right))
\]</span></p>
<p><span class="math display">\[
\beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\
0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\
0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\
1 & 1 \end{matrix}\right))
\]</span></p>
<p>n = 4000</p>
<pre class="r"><code>set.seed(1)
n = 4000; p = 2
Sigma = matrix(c(1,0.5,0.5,1),p,p)
U0 = matrix(0,2,2)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[2,2] = 1
U3 = matrix(1,2,2)
Utrue = list(U0=U0, U1=U1, U2=U2, U3=U3)
data = generate_data(n, p, Sigma, Utrue)</code></pre>
<pre class="r"><code>m.data = mash_set_data(data$Bhat, data$Shat)
U.c = cov_canonical(m.data)
grid = mashr:::autoselect_grid(m.data, sqrt(2))
Ulist = mashr:::normalize_Ulist(U.c)
xUlist = mashr:::expand_cov(Ulist,grid,usepointmass = TRUE)
result <- mixture.EM2.times(data$Bhat, xUlist, init_rho = c(-0.7,0,0.7), grid=1)
plot(result$result$log_liks)</code></pre>
<p><img src="figure/EstimateCorMaxEM2.Rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /> The estimated <span class="math inline">\(\rho\)</span> is 0.5576293.</p>
<pre class="r"><code>m.data.em = mash_set_data(data$Bhat, data$Shat, V = matrix(c(1,result[[1]]$rho,result[[1]]$rho,1),2,2))
U.c = cov_canonical(m.data.em)
m.em = mash(m.data.em, U.c, verbose= FALSE)
null.ind = which(apply(data$B,1,sum) == 0)</code></pre>
<p>The log likelihood is -1.23021110^{4}. There are 37 significant samples, 1 false positives. The RRMSE is 0.5871084.</p>
<p>The estimated <code>pi</code> is</p>
<pre class="r"><code>barplot(get_estimated_pi(m.em), las=2, cex.names = 0.7, main='EM rho', ylim=c(0,0.8))</code></pre>
<p><img src="figure/EstimateCorMaxEM2.Rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>The ROC curve:</p>
<pre class="r"><code>m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.correct = mash(m.data.correct, U.c, verbose = FALSE)
m.correct.seq = ROC.table(data$B, m.correct)
m.em.seq = ROC.table(data$B, m.em)</code></pre>
<p><img src="figure/EstimateCorMaxEM2.Rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="session-information" class="section level1">
<h1>Session information</h1>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] kableExtra_0.9.0 knitr_1.20 mashr_0.2-11 ashr_2.2-10
loaded via a namespace (and not attached):
[1] Rcpp_0.12.18 compiler_3.5.1 pillar_1.3.0
[4] plyr_1.8.4 iterators_1.0.10 tools_3.5.1
[7] digest_0.6.15 viridisLite_0.3.0 evaluate_0.11
[10] tibble_1.4.2 lattice_0.20-35 pkgconfig_2.0.1
[13] rlang_0.2.1 Matrix_1.2-14 foreach_1.4.4
[16] rstudioapi_0.7 yaml_2.2.0 parallel_3.5.1
[19] mvtnorm_1.0-8 xml2_1.2.0 httr_1.3.1
[22] stringr_1.3.1 REBayes_1.3 hms_0.4.2
[25] rprojroot_1.3-2 grid_3.5.1 R6_2.2.2
[28] rmarkdown_1.10 rmeta_3.0 readr_1.1.1
[31] magrittr_1.5 scales_0.5.0 backports_1.1.2
[34] codetools_0.2-15 htmltools_0.3.6 MASS_7.3-50
[37] rvest_0.3.2 assertthat_0.2.0 colorspace_1.3-2
[40] stringi_1.2.4 Rmosek_8.0.69 munsell_0.5.0
[43] pscl_1.5.2 doParallel_1.0.11 truncnorm_1.0-8
[46] SQUAREM_2017.10-1 crayon_1.3.4 </code></pre>
</div>
<!-- Adjust MathJax settings so that all math formulae are shown using
TeX fonts only; see
http://docs.mathjax.org/en/latest/configuration.html. This will make
the presentation more consistent at the cost of the webpage sometimes
taking slightly longer to load. Note that this only works because the
footer is added to webpages before the MathJax javascript. -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
"HTML-CSS": { availableFonts: ["TeX"] }
});
</script>
<hr>
<p>
This <a href="http://rmarkdown.rstudio.com">R Markdown</a> site was created with <a href="https://github.com/jdblischak/workflowr">workflowr</a>
</p>
<hr>
<!-- To enable disqus, uncomment the section below and provide your disqus_shortname -->
<!-- disqus
<div id="disqus_thread"></div>
<script type="text/javascript">
/* * * CONFIGURATION VARIABLES: EDIT BEFORE PASTING INTO YOUR WEBPAGE * * */
var disqus_shortname = 'rmarkdown'; // required: replace example with your forum shortname
/* * * DON'T EDIT BELOW THIS LINE * * */
(function() {
var dsq = document.createElement('script'); dsq.type = 'text/javascript'; dsq.async = true;
dsq.src = '//' + disqus_shortname + '.disqus.com/embed.js';
(document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq);
})();
</script>
<noscript>Please enable JavaScript to view the <a href="http://disqus.com/?ref_noscript">comments powered by Disqus.</a></noscript>
<a href="http://disqus.com" class="dsq-brlink">comments powered by <span class="logo-disqus">Disqus</span></a>
-->
</div>
</div>
</div>
<script>
// add bootstrap table styles to pandoc tables
function bootstrapStylePandocTables() {
$('tr.header').parent('thead').parent('table').addClass('table table-condensed');
}
$(document).ready(function () {
bootstrapStylePandocTables();
});
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement("script");
script.type = "text/javascript";
script.src = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML";
document.getElementsByTagName("head")[0].appendChild(script);
})();
</script>
</body>
</html>