<!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="Lei Sun" />
<meta name="date" content="2018-02-05" />
<title>Knockoff on Small Signals</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>
<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-4.5.0/css/font-awesome.min.css" rel="stylesheet" />
<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;
}
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 -->
<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">truncash</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/LSun/truncash">
<span class="fa fa-github"></span>
</a>
</li>
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container -->
</div><!--/.navbar -->
<div class="fluid-row" id="header">
<h1 class="title toc-ignore"><code>Knockoff</code> on Small Signals</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-02-05</em></h4>
</div>
<!-- The file analysis/chunks.R contains chunks that define default settings
shared across the workflowr files. -->
<!-- Update knitr chunk options -->
<!-- Insert the date the file was last updated -->
<p><strong>Last updated:</strong> 2018-02-15</p>
<!-- Insert the code version (Git commit SHA1) if Git repository exists and R
package git2r is installed -->
<p><strong>Code version:</strong> c586cb3</p>
<!-- Add your analysis here -->
<section id="introduction" class="level2">
<h2>Introduction</h2>
<p>In the <code>Knockoff</code> paper simulations, <span class="math inline">\(\beta\)</span>’s are either <span class="math inline">\(0\)</span> or <span class="math inline">\(A\)</span>. Here we are replicating the results, and investigating how well <code>Knockoff</code> deal with small signals.</p>
<p>In the following simulations, we always have <span class="math inline">\(n = 3000\)</span>, <span class="math inline">\(p = 1000\)</span>, For a certain <span class="math inline">\(\beta\)</span>, <span class="math inline">\(Y_n \sim N(X_{n\times p}\beta_p, I_n)\)</span>. Out of <span class="math inline">\(p = 1000\)</span> <span class="math inline">\(\beta_j\)</span>’s, here are three scenarios.</p>
<ul>
<li>Scenario 1: <span class="math inline">\(950\)</span> are 0, and the rest <span class="math inline">\(50\)</span> are <span class="math inline">\(A = 3.5\)</span>. (replicating a data point on Fig 3 of the <code>Knockoff</code> paper)</li>
<li>Scenario 2: <span class="math inline">\(850\)</span> are 0, <span class="math inline">\(50\)</span> are <span class="math inline">\(A = 3.5\)</span> as large signals, and the rest <span class="math inline">\(100\)</span> are small signals uniformly from <span class="math inline">\(0\)</span> to <span class="math inline">\(3.5\)</span>.</li>
<li>Scenario 3: <span class="math inline">\(750\)</span> are 0, <span class="math inline">\(50\)</span> are <span class="math inline">\(A = 3.5\)</span> as large signals, and the rest <span class="math inline">\(200\)</span> are small signals uniformly from <span class="math inline">\(0\)</span> to <span class="math inline">\(3.5\)</span>.</li>
</ul>
<pre class="r"><code>n <- 3000
p <- 1000
k <- 50
q <- 0.1
A <- 3.5</code></pre>
<section id="scenario-1-50-large-signals-no-small-signals-950-zeroes." class="level3">
<h3>Scenario 1: 50 large signals, no small signals, 950 zeroes.</h3>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- svd(X)$u
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="scenario-2-50-large-signals-100-small-signals-850-zeroes." class="level3">
<h3>Scenario 2: 50 large signals, 100 small signals, 850 zeroes.</h3>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- svd(X)$u
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="scenario-3-50-large-signals-200-small-signals-750-zeroes." class="level3">
<h3>Scenario 3: 50 large signals, 200 small signals, 750 zeroes.</h3>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- svd(X)$u
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-15-1.png" width="672" style="display: block; margin: auto;" /></p>
</section>
</section>
<section id="orthogonal-design" class="level2">
<h2>Orthogonal design</h2>
<p><span class="math inline">\(X\)</span> has random orthonormal columns</p>
<table>
<thead>
<tr class="header">
<th style="text-align: right;">FDP.BH</th>
<th style="text-align: right;">FDP.Knockoff</th>
<th style="text-align: right;">FDP.Knockoff.Plus</th>
<th style="text-align: right;">Power.BH</th>
<th style="text-align: right;">Power.Knockoff</th>
<th style="text-align: right;">Power.Knockoff.Plus</th>
<th style="text-align: right;">Power.Large.BH</th>
<th style="text-align: right;">Power.Large.Knockoff</th>
<th style="text-align: right;">Power.Large.Knockoff.Plus</th>
<th style="text-align: right;">Power.Small.BH</th>
<th style="text-align: right;">Power.Small.Knockoff</th>
<th style="text-align: right;">Power.Small.Knockoff.Plus</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.0979</td>
<td style="text-align: right;">0.1082</td>
<td style="text-align: right;">0.0863</td>
<td style="text-align: right;">0.7339</td>
<td style="text-align: right;">0.7329</td>
<td style="text-align: right;">0.6939</td>
<td style="text-align: right;">0.7339</td>
<td style="text-align: right;">0.7329</td>
<td style="text-align: right;">0.6939</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">NA</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.0847</td>
<td style="text-align: right;">0.0921</td>
<td style="text-align: right;">0.0795</td>
<td style="text-align: right;">0.3836</td>
<td style="text-align: right;">0.3867</td>
<td style="text-align: right;">0.3691</td>
<td style="text-align: right;">0.7791</td>
<td style="text-align: right;">0.7799</td>
<td style="text-align: right;">0.7555</td>
<td style="text-align: right;">0.1858</td>
<td style="text-align: right;">0.1901</td>
<td style="text-align: right;">0.176</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.0756</td>
<td style="text-align: right;">0.0817</td>
<td style="text-align: right;">0.0727</td>
<td style="text-align: right;">0.3294</td>
<td style="text-align: right;">0.3340</td>
<td style="text-align: right;">0.3195</td>
<td style="text-align: right;">0.8116</td>
<td style="text-align: right;">0.8122</td>
<td style="text-align: right;">0.7932</td>
<td style="text-align: right;">0.2088</td>
<td style="text-align: right;">0.2145</td>
<td style="text-align: right;">0.201</td>
</tr>
</tbody>
</table>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-19-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-20-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-21-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-22-1.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="independent-design" class="level2">
<h2>Independent design</h2>
<p><span class="math inline">\(X_{n \times p}\)</span> has independent columns simulated from <span class="math inline">\(N(0, 1)\)</span> and then normalized to have <span class="math inline">\(\|X_j\|_2^2 \equiv 1\)</span>.</p>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<table>
<thead>
<tr class="header">
<th style="text-align: right;">FDP.BH</th>
<th style="text-align: right;">FDP.Knockoff</th>
<th style="text-align: right;">FDP.Knockoff.Plus</th>
<th style="text-align: right;">Power.BH</th>
<th style="text-align: right;">Power.Knockoff</th>
<th style="text-align: right;">Power.Knockoff.Plus</th>
<th style="text-align: right;">Power.Large.BH</th>
<th style="text-align: right;">Power.Large.Knockoff</th>
<th style="text-align: right;">Power.Large.Knockoff.Plus</th>
<th style="text-align: right;">Power.Small.BH</th>
<th style="text-align: right;">Power.Small.Knockoff</th>
<th style="text-align: right;">Power.Small.Knockoff.Plus</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.0950</td>
<td style="text-align: right;">0.0713</td>
<td style="text-align: right;">0.0483</td>
<td style="text-align: right;">0.4344</td>
<td style="text-align: right;">0.5399</td>
<td style="text-align: right;">0.4135</td>
<td style="text-align: right;">0.4344</td>
<td style="text-align: right;">0.5399</td>
<td style="text-align: right;">0.4135</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">NA</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.0828</td>
<td style="text-align: right;">0.0587</td>
<td style="text-align: right;">0.0397</td>
<td style="text-align: right;">0.2180</td>
<td style="text-align: right;">0.2106</td>
<td style="text-align: right;">0.1584</td>
<td style="text-align: right;">0.4789</td>
<td style="text-align: right;">0.4820</td>
<td style="text-align: right;">0.3699</td>
<td style="text-align: right;">0.0876</td>
<td style="text-align: right;">0.0749</td>
<td style="text-align: right;">0.0527</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.0747</td>
<td style="text-align: right;">0.0456</td>
<td style="text-align: right;">0.0325</td>
<td style="text-align: right;">0.1838</td>
<td style="text-align: right;">0.1613</td>
<td style="text-align: right;">0.1243</td>
<td style="text-align: right;">0.5135</td>
<td style="text-align: right;">0.4372</td>
<td style="text-align: right;">0.3432</td>
<td style="text-align: right;">0.1013</td>
<td style="text-align: right;">0.0923</td>
<td style="text-align: right;">0.0696</td>
</tr>
</tbody>
</table>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-36-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-37-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-38-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-39-1.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="local-correlation-design" class="level2">
<h2>Local correlation design</h2>
<p><span class="math inline">\(X_{n \times p}\)</span> has correlation <span class="math inline">\(\Sigma_{ij} = \rho^{|i - j|}\)</span>. Each row is independently <span class="math inline">\(N(0, \Sigma)\)</span> and then normalized to have <span class="math inline">\(\|X_j\|_2^2 \equiv 1\)</span>.</p>
<pre class="r"><code>rho <- 0.25
Sigma <- toeplitz(rho^(0 : (p - 1)))
X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
X <- X %*% chol(Sigma)
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
X <- X %*% chol(Sigma)
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<pre class="r"><code>X <- matrix(rnorm(n * p), n , p)
X <- t(t(X) / sqrt(colSums(X^2)))
X <- X %*% chol(Sigma)
Xk <- knockoff::create.fixed(X)
Xk <- Xk$Xk</code></pre>
<table>
<thead>
<tr class="header">
<th style="text-align: right;">FDP.BH</th>
<th style="text-align: right;">FDP.Knockoff</th>
<th style="text-align: right;">FDP.Knockoff.Plus</th>
<th style="text-align: right;">Power.BH</th>
<th style="text-align: right;">Power.Knockoff</th>
<th style="text-align: right;">Power.Knockoff.Plus</th>
<th style="text-align: right;">Power.Large.BH</th>
<th style="text-align: right;">Power.Large.Knockoff</th>
<th style="text-align: right;">Power.Large.Knockoff.Plus</th>
<th style="text-align: right;">Power.Small.BH</th>
<th style="text-align: right;">Power.Small.Knockoff</th>
<th style="text-align: right;">Power.Small.Knockoff.Plus</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">0.0957</td>
<td style="text-align: right;">0.0563</td>
<td style="text-align: right;">0.0344</td>
<td style="text-align: right;">0.3367</td>
<td style="text-align: right;">0.4985</td>
<td style="text-align: right;">0.3484</td>
<td style="text-align: right;">0.3367</td>
<td style="text-align: right;">0.4985</td>
<td style="text-align: right;">0.3484</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">NA</td>
<td style="text-align: right;">NA</td>
</tr>
<tr class="even">
<td style="text-align: right;">0.0824</td>
<td style="text-align: right;">0.0407</td>
<td style="text-align: right;">0.0249</td>
<td style="text-align: right;">0.1710</td>
<td style="text-align: right;">0.1656</td>
<td style="text-align: right;">0.1070</td>
<td style="text-align: right;">0.3820</td>
<td style="text-align: right;">0.3620</td>
<td style="text-align: right;">0.2348</td>
<td style="text-align: right;">0.0655</td>
<td style="text-align: right;">0.0673</td>
<td style="text-align: right;">0.0431</td>
</tr>
<tr class="odd">
<td style="text-align: right;">0.0738</td>
<td style="text-align: right;">0.0258</td>
<td style="text-align: right;">0.0170</td>
<td style="text-align: right;">0.1441</td>
<td style="text-align: right;">0.1393</td>
<td style="text-align: right;">0.0988</td>
<td style="text-align: right;">0.4137</td>
<td style="text-align: right;">0.4040</td>
<td style="text-align: right;">0.2893</td>
<td style="text-align: right;">0.0767</td>
<td style="text-align: right;">0.0731</td>
<td style="text-align: right;">0.0511</td>
</tr>
</tbody>
</table>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-53-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-54-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-55-1.png" width="672" style="display: block; margin: auto;" /></p>
<p><img src="figure/knockoff_6.rmd/unnamed-chunk-56-1.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="session-information" class="level2">
<h2>Session information</h2>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.3
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] knitr_1.19 ggplot2_2.2.1 knockoff_0.3.0
loaded via a namespace (and not attached):
[1] Rcpp_0.12.14 magrittr_1.5 munsell_0.4.3 colorspace_1.3-2
[5] rlang_0.1.6 highr_0.6 stringr_1.2.0 plyr_1.8.4
[9] tools_3.4.3 grid_3.4.3 gtable_0.2.0 git2r_0.21.0
[13] htmltools_0.3.6 yaml_2.1.16 lazyeval_0.2.1 rprojroot_1.3-2
[17] digest_0.6.14 tibble_1.4.1 evaluate_0.10.1 rmarkdown_1.8
[21] labeling_0.3 stringi_1.1.6 compiler_3.4.3 pillar_1.0.1
[25] scales_0.5.0 backports_1.1.2 </code></pre>
</section>
<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>