<!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="Joyce Hsiao" />


<title>Evaluation of cellcycleR on Leng data</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">fucci-seq</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/fucci-seq">
    <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">Evaluation of cellcycleR on Leng data</h1>
<h4 class="author"><em>Joyce Hsiao</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> 2018-02-09</p>
<!-- Insert the code version (Git commit SHA1) if Git repository exists and R
 package git2r is installed -->
<p><strong>Code version:</strong> 5845152</p>
<hr />
<div id="overviewresults" class="section level2">
<h2>Overview/Results</h2>
<p>Goal: Evalute the performance on cellcycleR on RNA-seq data</p>
<p>Method: Apply cellcycleR to Leng et al. 2015 data. Run each model three times on different random seeds. The number of discrete time intervals is fixed to be 100 between 0 to <span class="math inline">\(2\pi\)</span>.</p>
<p>Data: We normalize the expression counts by library size and convert the expression counts to log2CPM.</p>
<p>Results:</p>
<pre><code>1. Cell time estimates: Should span 0 to 2pi give that the samples collected including somewhat similar number of cells from each phase. This was observed for results using Oscope genes (29) but not for results using all genes or cell cycle annotated genes.

2. Cell order: We exepct that at least the ordering of cells by cell time should match with the fucci-labeled cell cycle phases. This was the case for results using Oscope genes (29) but not for results using all genes or cell cycle annotated genes. </code></pre>
<p>Additional analysis 1: I was thinking that perhaps cellcycleR doesn’t perform well when the number of cells in the different phases is uneven. And this could explain the poor performance on the fucci data. Or from the method perspective, it could be that the model doesn’t accommodate skewed/distored oscillation patterns in the data. I then created a dataset from the Leng data where there’s only 10 cells in G1. Results show that (at the end of this page) most G1 cells are estimated to be similar to G2 cells according to the estimated cell ordering…</p>
<p>Additional analysis 2: I examined the cellcycleR fit on the whole set of Leng data gene. I found that the ones with low signal-to-noise ratio range from genes with high variance-mean dependency to low variance-mean dependency.</p>
<hr />
</div>
<div id="data-and-packages" class="section level2">
<h2>Data and packages</h2>
<p>Packages</p>
<pre class="r"><code>library(Biobase)
library(ggplot2)
library(cowplot)
library(data.table)
library(tidyr)
library(gplots)
library(cellcycleR)
library(VennDiagram)
library(ccRemover)
library(scales)
library(circular)
library(limma)</code></pre>
<p>Leng et al. data</p>
<pre class="r"><code>HumanLengESC &lt;- readRDS(file = &quot;../data/rnaseq-previous-studies/HumanLengESC.rds&quot;)

#select fucci-expression cells
counts_leng &lt;- exprs(HumanLengESC)[,pData(HumanLengESC)$cell_state != &quot;H1&quot;]
libsize_leng &lt;- colSums(counts_leng)
pdata_leng &lt;- pData(HumanLengESC)[pData(HumanLengESC)$cell_state != &quot;H1&quot;, ]
cpm_leng &lt;- (t(t(counts_leng)*(10^6)/libsize_leng))[,pdata_leng$cell_state != &quot;H1&quot;]
log2cpm_leng &lt;- (log2(cpm_leng+1))[,pdata_leng$cell_state != &quot;H1&quot;]
pdata_leng$cell_state &lt;- droplevels(pdata_leng$cell_state)

table(pdata_leng$cell_state)</code></pre>
<pre><code>
G1 G2  S 
91 76 80 </code></pre>
<pre class="r"><code>## filter genes
genes_to_include &lt;- which(rowMeans(cpm_leng)&gt;1)
log2cpm_leng &lt;- log2cpm_leng[genes_to_include,]

## subset to include genes that are annotated as cell cycle genes (according to ccRemover)
ccremover &lt;- readRDS(&quot;../data/cellcycle-genes-previous-studies/rds/macosko-2015.rds&quot;)
which_ccremover &lt;- gene_indexer(rownames(log2cpm_leng), species=&quot;human&quot;, name_type=&quot;symbol&quot;)

log2cpm_leng_ccremover &lt;- log2cpm_leng[which_ccremover, ]
cpm_leng_ccremover &lt;- cpm_leng[which_ccremover, ]
counts_leng_ccremover &lt;- counts_leng[which_ccremover, ]

## use Oscope 29 genes
oscope &lt;- readRDS(&quot;../data/cellcycle-genes-previous-studies/rds/leng-2015.rds&quot;)

log2cpm_leng_oscope &lt;- log2cpm_leng[which(rownames(log2cpm_leng) %in% unique(oscope$hgnc)),]
counts_leng_oscope &lt;- counts_leng[which(rownames(counts_leng) %in% unique(oscope$hgnc)),]


## standardize log2cpm for fitting cellcycleR
log2cpm_leng_oscope_z &lt;- t(apply(log2cpm_leng_oscope, 1, scale))
colnames(log2cpm_leng_oscope_z) &lt;- colnames(log2cpm_leng_oscope)

log2cpm_leng_ccremover_z &lt;- t(apply(log2cpm_leng_ccremover, 1, scale))
colnames(log2cpm_leng_ccremover_z) &lt;- colnames(log2cpm_leng_ccremover)

log2cpm_leng_z &lt;- t(apply(log2cpm_leng, 1, scale))
colnames(log2cpm_leng_z) &lt;- colnames(log2cpm_leng)</code></pre>
<p>compare the sets</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
<hr />
</div>
<div id="main-analysis" class="section level2">
<h2>Main analysis</h2>
<p>Fitting cellcycleR using the same 3 random seeds for each analysis.</p>
<pre class="r"><code>seeds &lt;- c(101,231,456)

fit_leng_list &lt;- vector(&quot;list&quot;, length(seeds))
fit_leng_ccremover_list &lt;- vector(&quot;list&quot;, length(seeds))
fit_leng_oscope_list &lt;- vector(&quot;list&quot;, length(seeds))

for (i in 1:length(seeds)) {
  set.seed(seeds[i])
  fit_leng_list[[i]] &lt;- sin_cell_ordering_class(t(log2cpm_leng_z), 
                                           celltime_levels=100, 
                                           num_shuffle=1, maxiter=300, tol=1e-6,
                                           fix.phase = FALSE, phase_in=NULL,
                                           n_cores=8)
  set.seed(seeds[i])
  fit_leng_ccremover_list[[i]] &lt;- sin_cell_ordering_class(t(log2cpm_leng_ccremover_z), 
                                           celltime_levels=100, 
                                           num_shuffle=1, maxiter=300, tol=1e-6,
                                           fix.phase = FALSE, phase_in=NULL,
                                           n_cores=8)
  set.seed(seeds[i])
  fit_leng_oscope_list[[i]] &lt;- sin_cell_ordering_class(t(log2cpm_leng_oscope_z), 
                                           celltime_levels=100, 
                                           num_shuffle=1, maxiter=300, tol=1e-6,
                                           fix.phase = FALSE, phase_in=NULL,
                                           n_cores=8)
}

saveRDS(fit_leng_list, &quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng.rds&quot;) 
saveRDS(fit_leng_ccremover_list, &quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng_ccremover.rds&quot;) 
saveRDS(fit_leng_oscope_list, &quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng_oscope.rds&quot;) </code></pre>
<p>Load previously computed results.</p>
<pre class="r"><code>fit_leng_list &lt;- readRDS(&quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng.rds&quot;) 
fit_leng_ccremover_list &lt;- readRDS(&quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng_ccremover.rds&quot;) 
fit_leng_oscope_list &lt;- readRDS(&quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng_oscope.rds&quot;) 

fit_leng &lt;- fit_leng_list[[which.max(sapply(fit_leng_list, &quot;[[&quot;, &quot;loglik&quot;))]]
fit_leng_ccremover &lt;- fit_leng_ccremover_list[[which.max(sapply(fit_leng_ccremover_list, &quot;[[&quot;, &quot;loglik&quot;))]]
fit_leng_oscope &lt;- fit_leng_oscope_list[[which.max(sapply(fit_leng_oscope_list, &quot;[[&quot;, &quot;loglik&quot;))]]</code></pre>
</div>
<div id="results" class="section level2">
<h2>Results</h2>
<p>Distribution of cell times</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-6-1.png" width="768" style="display: block; margin: auto;" /></p>
<p>Estimated cell time and FUCCI labels</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>Estimated cell time and sorted phase</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>Results on the same data. Visualize on circular plot.</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-9-1.png" width="768" style="display: block; margin: auto;" /></p>
<p><strong>gene profiles</strong></p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
<hr />
</div>
<div id="subsampling" class="section level2">
<h2>Subsampling</h2>
<p>Goal: To test the fit of cellcycleR on dataset with small proportion of G1 cells.</p>
<p>Method: Here I am using the Oscope genes to see if cellcycleR still performs well with a small dataset.</p>
<pre class="r"><code>seeds &lt;- c(101,231,456)
fit_leng_oscope_short_list &lt;- vector(&quot;list&quot;, length(seeds))

pdata_leng$cell_state &lt;- as.character(pdata_leng$cell_state)

g1_cells &lt;- which(as.character(pdata_leng$cell_state) == &quot;G1&quot;)
which_subsample &lt;- g1_cells[sample(length(g1_cells), 81)]
log2cpm_leng_oscope_short &lt;- log2cpm_leng_oscope[,-which_subsample]
pdata_leng_short &lt;- pdata_leng[-which_subsample,]

log2cpm_leng_oscope_short_z &lt;- t(apply(log2cpm_leng_oscope_short, 1, scale))
colnames(log2cpm_leng_oscope_short_z) &lt;- colnames(log2cpm_leng_oscope_short)

for (i in 1:length(seeds)) {
  set.seed(seeds[i])
  fit_leng_oscope_short_list[[i]] &lt;- sin_cell_ordering_class(t(log2cpm_leng_oscope_short_z), 
                                           celltime_levels=100, 
                                           num_shuffle=1, maxiter=300, tol=1e-6,
                                           fix.phase = FALSE, phase_in=NULL,
                                           n_cores=8)
}

saveRDS(fit_leng_oscope_short_list, &quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng_oscope_short.rds&quot;) 
saveRDS(pdata_leng_short, &quot;../output/cellcycler-seqdata-leng.Rmd/pdata_leng_short.rds&quot;) </code></pre>
<p>Load previously computed results.</p>
<pre class="r"><code>fit_leng_oscope_short_list &lt;- readRDS(&quot;../output/cellcycler-seqdata-leng.Rmd/fit_leng_oscope_short.rds&quot;) 
pdata_leng_short &lt;- readRDS(&quot;../output/cellcycler-seqdata-leng.Rmd/pdata_leng_short.rds&quot;) 

fit_leng_oscope_short &lt;- fit_leng_oscope_short_list[[which.max(sapply(fit_leng_oscope_short_list, &quot;[[&quot;, &quot;loglik&quot;))]]</code></pre>
<p>Results</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-13-1.png" width="768" style="display: block; margin: auto;" /></p>
<p>Results on the same data. Visualize on circular plot.</p>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-14-1.png" width="768" style="display: block; margin: auto;" /></p>
<hr />
</div>
<div id="overdispersion" class="section level2">
<h2>Overdispersion</h2>
<p>Let’s see if genes with high SNR are also those with high overdispersion. Use voom because we are interested in log based models.</p>
<pre class="r"><code>par(mfrow=c(1,2))
leng_voom &lt;- voom(counts_leng, lib.size=libsize_leng, normalize.method = &quot;none&quot;,
                  save.plot = TRUE)
genes_voom &lt;- which(rownames(counts_leng) %in% names(leng_voom$voom.xy$x))
snr &lt;- with(fit_leng, (amp[genes_voom]^2)/(sigma[genes_voom]^2))
plot(leng_voom$voom.xy$x, leng_voom$voom.xy$y, pch=16, cex=.5, col = &quot;gray60&quot;,
     xlab = &quot;log2(count sizse + .5&quot;, ylab=&quot;Sqrt(standard deviation)&quot;,
     main = &quot;All genes&quot;)
lines(leng_voom$voom.line, col=&quot;red&quot;)
points(cbind(leng_voom$voom.xy$x,leng_voom$voom.xy$y)[order(snr, decreasing=F)[1:50],],
       pch=1, col=&quot;blue&quot;)


leng_voom &lt;- voom(counts_leng_ccremover, lib.size=libsize_leng, normalize.method = &quot;none&quot;,
                  save.plot = TRUE)
genes_voom &lt;- which(rownames(counts_leng_ccremover) %in% names(leng_voom$voom.xy$x))
snr &lt;- with(fit_leng_ccremover, (amp[genes_voom]^2)/(sigma[genes_voom]^2))
plot(leng_voom$voom.xy$x, leng_voom$voom.xy$y, pch=16, cex=.5, col = &quot;gray60&quot;,
     xlab = &quot;log2(count sizse + .5&quot;, ylab=&quot;Sqrt(standard deviation)&quot;,
     main = &quot;All genes&quot;)
lines(leng_voom$voom.line, col=&quot;red&quot;)
points(cbind(leng_voom$voom.xy$x,leng_voom$voom.xy$y)[order(snr, decreasing=F)[1:50],],
       pch=1, col=&quot;blue&quot;)</code></pre>
<p><img src="figure/cellcycler-seqdata-leng.Rmd/unnamed-chunk-15-1.png" width="672" style="display: block; margin: auto;" /></p>
<hr />
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre><code>R version 3.4.1 (2017-06-30)
Platform: x86_64-redhat-linux-gnu (64-bit)
Running under: Scientific Linux 7.2 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /usr/lib64/R/lib/libRblas.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] grid      parallel  stats     graphics  grDevices utils     datasets 
[8] methods   base     

other attached packages:
 [1] limma_3.34.8        circular_0.4-93     scales_0.5.0       
 [4] ccRemover_1.0.4     VennDiagram_1.6.18  futile.logger_1.4.3
 [7] cellcycleR_0.1.6    zoo_1.8-1           binhf_1.0-1        
[10] adlift_1.3-3        EbayesThresh_1.4-12 wavethresh_4.6.8   
[13] MASS_7.3-47         gplots_3.0.1        tidyr_0.8.0        
[16] data.table_1.10.4-3 cowplot_0.9.2       ggplot2_2.2.1      
[19] Biobase_2.38.0      BiocGenerics_0.24.0

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.15         pillar_1.1.0         compiler_3.4.1      
 [4] git2r_0.21.0         plyr_1.8.4           futile.options_1.0.0
 [7] bitops_1.0-6         tools_3.4.1          boot_1.3-19         
[10] digest_0.6.15        lattice_0.20-35      evaluate_0.10.1     
[13] tibble_1.4.2         gtable_0.2.0         rlang_0.1.6         
[16] yaml_2.1.16          mvtnorm_1.0-7        stringr_1.2.0       
[19] knitr_1.19           gtools_3.5.0         caTools_1.17.1      
[22] rprojroot_1.3-2      glue_1.2.0           rmarkdown_1.8       
[25] gdata_2.18.0         lambda.r_1.2         purrr_0.2.4         
[28] magrittr_1.5         backports_1.1.2      htmltools_0.3.6     
[31] colorspace_1.3-2     KernSmooth_2.23-15   stringi_1.1.6       
[34] lazyeval_0.2.1       munsell_0.4.3       </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>