<!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="2017-03-26" />
<title>Empirical Null with Gaussian Derivatives: Numerical Issues</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 -->
<!-- Add a small amount of space between sections. -->
<style type="text/css">
div.section {
padding-top: 12px;
}
</style>
<div class="fluid-row" id="header">
<h1 class="title toc-ignore">Empirical Null with Gaussian Derivatives: Numerical Issues</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-03-26</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-05-12</p>
<strong>workflowr checks:</strong> <small>(Click a bullet for more information)</small>
<ul>
<li>
<details>
<p><summary> <strong style="color:blue;">✔</strong> <strong>R Markdown file:</strong> up-to-date </summary></p>
<p>Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.</p>
</details>
</li>
<li>
<details>
<p><summary> <strong style="color:blue;">✔</strong> <strong>Environment:</strong> empty </summary></p>
<p>Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.</p>
</details>
</li>
<li>
<details>
<p><summary> <strong style="color:blue;">✔</strong> <strong>Seed:</strong> <code>set.seed(12345)</code> </summary></p>
<p>The command <code>set.seed(12345)</code> was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.</p>
</details>
</li>
<li>
<details>
<p><summary> <strong style="color:blue;">✔</strong> <strong>Session information:</strong> recorded </summary></p>
<p>Great job! Recording the operating system, R version, and package versions is critical for reproducibility.</p>
</details>
</li>
<li>
<details>
<p><summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/LSun/truncash/tree/ddf906244e82e0675854c40da76dc16cf1a78fa4" target="_blank">ddf9062</a> </summary></p>
Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated. <br><br> Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use <code>wflow_publish</code> or <code>wflow_git_commit</code>). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
<pre><code>
Ignored files:
Ignored: .DS_Store
Ignored: .Rhistory
Ignored: .Rproj.user/
Ignored: analysis/.DS_Store
Ignored: analysis/BH_robustness_cache/
Ignored: analysis/FDR_Null_cache/
Ignored: analysis/FDR_null_betahat_cache/
Ignored: analysis/Rmosek_cache/
Ignored: analysis/StepDown_cache/
Ignored: analysis/alternative2_cache/
Ignored: analysis/alternative_cache/
Ignored: analysis/ash_gd_cache/
Ignored: analysis/average_cor_gtex_2_cache/
Ignored: analysis/average_cor_gtex_cache/
Ignored: analysis/brca_cache/
Ignored: analysis/cash_deconv_cache/
Ignored: analysis/cash_fdr_1_cache/
Ignored: analysis/cash_fdr_2_cache/
Ignored: analysis/cash_fdr_3_cache/
Ignored: analysis/cash_fdr_4_cache/
Ignored: analysis/cash_fdr_5_cache/
Ignored: analysis/cash_fdr_6_cache/
Ignored: analysis/cash_plots_cache/
Ignored: analysis/cash_sim_1_cache/
Ignored: analysis/cash_sim_2_cache/
Ignored: analysis/cash_sim_3_cache/
Ignored: analysis/cash_sim_4_cache/
Ignored: analysis/cash_sim_5_cache/
Ignored: analysis/cash_sim_6_cache/
Ignored: analysis/cash_sim_7_cache/
Ignored: analysis/correlated_z_2_cache/
Ignored: analysis/correlated_z_3_cache/
Ignored: analysis/correlated_z_cache/
Ignored: analysis/create_null_cache/
Ignored: analysis/cutoff_null_cache/
Ignored: analysis/design_matrix_2_cache/
Ignored: analysis/design_matrix_cache/
Ignored: analysis/diagnostic_ash_cache/
Ignored: analysis/diagnostic_correlated_z_2_cache/
Ignored: analysis/diagnostic_correlated_z_3_cache/
Ignored: analysis/diagnostic_correlated_z_cache/
Ignored: analysis/diagnostic_plot_2_cache/
Ignored: analysis/diagnostic_plot_cache/
Ignored: analysis/efron_leukemia_cache/
Ignored: analysis/fitting_normal_cache/
Ignored: analysis/gaussian_derivatives_2_cache/
Ignored: analysis/gaussian_derivatives_3_cache/
Ignored: analysis/gaussian_derivatives_4_cache/
Ignored: analysis/gaussian_derivatives_5_cache/
Ignored: analysis/gaussian_derivatives_cache/
Ignored: analysis/gd-ash_cache/
Ignored: analysis/gd_delta_cache/
Ignored: analysis/gd_lik_2_cache/
Ignored: analysis/gd_lik_cache/
Ignored: analysis/gd_w_cache/
Ignored: analysis/knockoff_10_cache/
Ignored: analysis/knockoff_2_cache/
Ignored: analysis/knockoff_3_cache/
Ignored: analysis/knockoff_4_cache/
Ignored: analysis/knockoff_5_cache/
Ignored: analysis/knockoff_6_cache/
Ignored: analysis/knockoff_7_cache/
Ignored: analysis/knockoff_8_cache/
Ignored: analysis/knockoff_9_cache/
Ignored: analysis/knockoff_cache/
Ignored: analysis/knockoff_var_cache/
Ignored: analysis/marginal_z_alternative_cache/
Ignored: analysis/marginal_z_cache/
Ignored: analysis/mosek_reg_2_cache/
Ignored: analysis/mosek_reg_4_cache/
Ignored: analysis/mosek_reg_5_cache/
Ignored: analysis/mosek_reg_6_cache/
Ignored: analysis/mosek_reg_cache/
Ignored: analysis/pihat0_null_cache/
Ignored: analysis/plot_diagnostic_cache/
Ignored: analysis/poster_obayes17_cache/
Ignored: analysis/real_data_simulation_2_cache/
Ignored: analysis/real_data_simulation_3_cache/
Ignored: analysis/real_data_simulation_4_cache/
Ignored: analysis/real_data_simulation_5_cache/
Ignored: analysis/real_data_simulation_cache/
Ignored: analysis/rmosek_primal_dual_2_cache/
Ignored: analysis/rmosek_primal_dual_cache/
Ignored: analysis/seqgendiff_cache/
Ignored: analysis/simulated_correlated_null_2_cache/
Ignored: analysis/simulated_correlated_null_3_cache/
Ignored: analysis/simulated_correlated_null_cache/
Ignored: analysis/simulation_real_se_2_cache/
Ignored: analysis/simulation_real_se_cache/
Ignored: analysis/smemo_2_cache/
Ignored: data/LSI/
Ignored: docs/.DS_Store
Ignored: docs/figure/.DS_Store
Ignored: output/fig/
Unstaged changes:
Deleted: analysis/cash_plots_fdp.Rmd
</code></pre>
Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
</details>
</li>
</ul>
<details>
<summary> <small><strong>Expand here to see past versions:</strong></small> </summary>
<ul>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
File
</th>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
<th style="text-align:left;">
Message
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/cc0ab8379469bc3726f1508cd81e4ecd6fef1b1a/analysis/gaussian_derivatives_3.rmd" target="_blank">cc0ab83</a>
</td>
<td style="text-align:left;">
Lei Sun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
update
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/gaussian_derivatives_3.html" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/853a484bfacf347e109f6c8fb3ffaab5f4d6cc02/docs/gaussian_derivatives_3.html" target="_blank">853a484</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-11-07
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/1ea081a9eeb7fd3101271eeefe10ef8be9993622/docs/gaussian_derivatives_3.html" target="_blank">1ea081a</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
<td style="text-align:left;">
sites
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/c803b43a9255f39ecac06315f7e88db746dddfdf/analysis/gaussian_derivatives_3.rmd" target="_blank">c803b43</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-03-27
</td>
<td style="text-align:left;">
fitting and write-up
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/c803b43a9255f39ecac06315f7e88db746dddfdf/docs/gaussian_derivatives_3.html" target="_blank">c803b43</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-03-27
</td>
<td style="text-align:left;">
fitting and write-up
</td>
</tr>
</tbody>
</table>
</ul>
</details>
<hr />
<div id="introduction-instability-and-overfitting-k-can-be-too-large" class="section level2">
<h2>Introduction: instability and overfitting – <span class="math inline">\(K\)</span> can be too large</h2>
<p>Things can go wrong in choosing the optimal <span class="math inline">\(\hat K\)</span>. We are using <a href="gaussian_derivatives_2.html">an automated rule</a>, yet sometimes the objective (log-likelihood) could fail to pass the optimal criterion before the optimization becomes unstable.</p>
<p>Recall that we make <a href="gaussian_derivatives.html">two key assumptions</a> to make the problem tractable. <em>In place of the original second constraint of non-negativity, we use <span class="math inline">\(n\)</span> observed <span class="math inline">\(z\)</span> scores instead of all <span class="math inline">\(x\in\mathbb{R}\)</span>.</em> Therefore, when <span class="math inline">\(K\)</span> gets larger, and the higher order Gaussian derivatives involved get more complicated, it’s possible that the optimal solution will satisfy the non-negativity constraint for all <span class="math inline">\(n\)</span> observed <span class="math inline">\(z\)</span> scores, <strong>but not the whole real line.</strong> <em>This issue also happens to <a href="gaussian_derivatives_2.html">some well-behaved examples</a> if looked closely.</em></p>
<p>Meanwhile, sometimes an optimal <span class="math inline">\(\hat K\)</span> can be found according to <a href="gaussian_derivatives_2.html">the rule</a>, but it looks like overfitting. A <span class="math inline">\(K < \hat K\)</span> appears better.</p>
<p>Here we have two examples.</p>
<pre class="r"><code>source("../code/ecdfz.R")</code></pre>
<pre class="r"><code>z = read.table("../output/z_null_liver_777.txt")
p = read.table("../output/p_null_liver_777.txt")</code></pre>
<pre class="r"><code>library(ashr)
DataSet = c(522, 724)
res_DataSet = list()
for (i in 1:length(DataSet)) {
zscore = as.numeric(z[DataSet[i], ])
fit.ecdfz = ecdfz.optimal(zscore)
fit.ash = ash(zscore, 1, method = "fdr")
fit.ash.pi0 = get_pi0(fit.ash)
pvalue = as.numeric(p[DataSet[i], ])
fd.bh = sum(p.adjust(pvalue, method = "BH") <= 0.05)
res_DataSet[[i]] = list(DataSet = DataSet[i], fit.ecdfz = fit.ecdfz, fit.ash = fit.ash, fit.ash.pi0 = fit.ash.pi0, fd.bh = fd.bh, zscore = zscore, pvalue = pvalue)
}</code></pre>
<pre class="r"><code>library(EQL)
x.pt = seq(-5, 5, 0.01)
H.pt = sapply(1:15, EQL::hermite, x = x.pt)</code></pre>
</div>
<div id="example-1-numerical-instability-when-k-is-too-large" class="section level2">
<h2>Example 1: Numerical instability when <span class="math inline">\(K\)</span> is too large</h2>
<pre><code>Data Set 724 : Number of BH's False Discoveries: 79 ; ASH's pihat0 = 0.01606004</code></pre>
<p><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /></p>
<p>In this example <a href="gaussian_derivatives_2.html">the automated rule</a> fails to find an optimal <span class="math inline">\(\hat K\)</span>. Note that the fitted log-likelihood increased until seemingly reached a plateau, but didn’t quite make the cut. After that, as <span class="math inline">\(K\)</span> keeps getting larger, the optimization becomes unstable. The blue <span class="math inline">\(K = 14\)</span> line obviously breaks <a href="gaussian_derivatives.html">the non-negativity constraint</a> for <span class="math inline">\(x \neq z_i\)</span>, the <span class="math inline">\(n\)</span> observed <span class="math inline">\(z\)</span> scores.</p>
</div>
<div id="example-2-overfitting-when-k-is-larger-than-what-appears-necessary" class="section level2">
<h2>Example 2: Overfitting when <span class="math inline">\(K\)</span> is larger than what appears necessary</h2>
<pre><code>Data Set 522 : Number of BH's False Discoveries: 4 ; ASH's pihat0 = 0.02083846</code></pre>
<p><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /></p>
<p>In this example <a href="gaussian_derivatives_2.html">the automated rule</a> is able to find an optimal <span class="math inline">\(\hat K = 8\)</span>. However, the green <span class="math inline">\(K = 6\)</span> lines seems better visually. Their difference in the fitted log-likelihood is very small, although larger than what the rule requires.</p>
</div>
<div id="conclusion" class="section level2">
<h2>Conclusion</h2>
<p>Things can go very wrong when the number of fitted Gaussian derivatives <span class="math inline">\(K\)</span> is too large, and it implies that we cannot blindly fit ever growing <span class="math inline">\(K\)</span> and hope the fitted log-likelihood converges. On the other hand, the good news is oftentimes we can still reach a pattern of increasing log-likelihoods, which gives a reasonable <span class="math inline">\(K\)</span>, before the optimization becomes unstable, although it might be not the optimal <span class="math inline">\(K\)</span> we would find by the current log-likelihood ratio test motivated rule.</p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<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 10.13.4
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] CVXR_0.95 EQL_1.0-0 ttutils_1.0-1
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 knitr_1.20 whisker_0.3-2
[4] magrittr_1.5 workflowr_1.0.1 bit_1.1-12
[7] lattice_0.20-35 R6_2.2.2 stringr_1.3.0
[10] tools_3.4.3 grid_3.4.3 R.oo_1.21.0
[13] git2r_0.21.0 scs_1.1-1 htmltools_0.3.6
[16] bit64_0.9-7 yaml_2.1.18 rprojroot_1.3-2
[19] digest_0.6.15 gmp_0.5-13.1 Matrix_1.2-12
[22] ECOSolveR_0.4 R.utils_2.6.0 evaluate_0.10.1
[25] rmarkdown_1.9 stringi_1.1.6 Rmpfr_0.6-1
[28] compiler_3.4.3 backports_1.1.2 R.methodsS3_1.7.1</code></pre>
</div>
<hr>
<p>
</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>
-->
<!-- 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 reproducible <a href="http://rmarkdown.rstudio.com">R Markdown</a>
analysis was created with
<a href="https://github.com/jdblischak/workflowr">workflowr</a> 1.0.1
</p>
<hr>
</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>