<!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>Hierarchicagl Flash</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-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 -->
<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">Hierarchicagl Flash</h1>
<h4 class="author"><em>Yuxin Zou</em></h4>
<h4 class="date"><em>7/14/2018</em></h4>
</div>
<!-- Update knitr chunk options -->
<!-- Insert the date the file was last updated -->
<p><strong>Last updated:</strong> 2018-07-14</p>
<!-- Insert the code version (Git commit SHA1) if Git repository exists and R
package git2r is installed -->
<p><strong>Code version:</strong> 2c8172d</p>
<pre class="r"><code>library(flashr); library(ggplot2); library(reshape2)</code></pre>
<p>Plot function:</p>
<pre class="r"><code>plot_HP = function(title_name, mat, absolute = TRUE){
p1 = ggplot(data = mat, aes(Var2, Var1, fill = value))+geom_tile(color = "white")+
scale_fill_gradient2(low = "darkblue", high = "darkred", mid = "white",
midpoint = 0, space = "Lab") + labs(title = title_name, y = "samples", x = "variables") +
# scale_fill_gradient2(low = "white", high = "grey6", mid = "grey9", midpoint = 0.41, limit = c(0,0.82), space = "Lab") + labs(title = title_name, y = "samples", x = "variables") +
theme_minimal() + theme(legend.position="none",plot.title = element_text(size = 12.9, face = "bold"))
p1
}</code></pre>
<div id="model-outline" class="section level1">
<h1>Model Outline</h1>
<p>In flash model, we have <span class="math display">\[
Y_{n\times p} = L_{n \times k}F_{p\times k}^{T} + E_{n\times p}
\]</span> We could find some patterns of the data Y based on the factors in F. But the patterns depend on the loadings L. We run flash on L again <span class="math display">\[
L_{n\times k} = O_{n\times r} A_{k\times r}^{T} + R_{n\times k}
\]</span></p>
<p>From the factors A, we can check the dependence of factors in F.</p>
<p>I simulate Y with 5 conditions (p = 5). Some have equal effect in all conditions; some have equal effect in consition 3, 4 and 5, but stronger effect in condition 1 and 2 (all effects are in the same direction); some have effect only in the last condition.</p>
<ul>
<li><p>Summary from the simulation:</p>
<p>From the simulation, we see factors with loadings both positive or both negative, and also see cases where one loading is positive and the other is negative.</p></li>
<li><p>Question:</p>
<p>We interprete the factor as a pattern in the data <code>Y</code>. For example, in GTEx data, we say a factor with approximately equal weight for every tissue reflects that many eQTLs have similar effects accross all tissues. I doubt whether our interpretation is correct. <strong>The pattern in Y depends on L</strong>. If some rows of L have 2 large entries, then those samples depend on 2 factors. The pattern in <code>Y</code> is actually a linear combination of the 2 factors. Therefore, we perform factor analysis on <code>L</code> again. Using the new factors <code>A</code>, we check if there is a linear combination of factors <code>F</code>.</p>
<p>My question is that the actual pattern in <code>Y</code> is still depending on the new loading <code>O</code>. The rows of loading <code>O</code> could also have several large entries, which suggests that the pattern in <code>Y</code> depends on a combination of factors. I don’t have an example for this case yet. So I think we cannot say the factors summarize the patterns in the data.</p>
<p>The pattern in the data depends on loadings, and we don’t have any summary about the loadings.</p></li>
</ul>
</div>
<div id="simulation" class="section level1">
<h1>Simulation</h1>
<pre class="r"><code>flashdata = function(n, FF, LL, Lsd, err_sd){
k = ncol(FF)
p = nrow(FF)
LLL = matrix(0,0,k)
for(i in 1:nrow(LL)){
temp = matrix(rnorm(n * k, sd = Lsd), n, k)
direction = c(sign(temp[,1]))
temp2 = direction * abs(temp)
LLL = rbind(LLL, t(t(temp2) * LL[i,] ))
}
Ytrue = LLL %*% t(FF)
Y = Ytrue + matrix(rnorm(n*nrow(LL)*p, sd = err_sd), n*nrow(LL), p)
return(list(Y = Y, Ytrue = Ytrue))
}</code></pre>
<pre class="r"><code>set.seed(1)
FF = cbind(c(1,1,1,1,1), c(1,1,0,0,0), c(0,0,0,0,1))
LL = rbind(c(1,0,0), c(0,0,1), c(0.3,0.7,0))
data = flashdata(500, FF, LL, 4,0.5)</code></pre>
<div id="flash-model-loading-opposite-sign" class="section level2">
<h2>Flash model (loading opposite sign)</h2>
<p>Using greefy+backfit flash method:</p>
<pre class="r"><code>f1 = flash(data$Y, greedy = TRUE, backfit = TRUE)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code>fitting factor/loading 4</code></pre>
<pre class="r"><code>ff1 = flash_get_ldf(f1)$f; Loadings1 = flash_get_ldf(f1)$l * flash_get_ldf(f1)$d</code></pre>
<p>The factors are</p>
<pre class="r"><code>par(mfrow=c(1,3))
for(i in 1:ncol(ff1)){
barplot(ff1[,i], main = paste0('Factor ', i, ', pve=', round(flash_get_pve(f1)[i],3)))
}</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>The factor 1 and 3 reflect the pattern in the simulated data. But for factor 2, we expect they have effect in the same direction. Factor 2 is not a pattern in the data. The real pattern is a combination of factor 1 and 2.</p>
<p>Let’s check the L matrix</p>
<pre class="r"><code>melted_Loading = melt(Loadings1)
melted_Loading_zoom = melt(Loadings1[1001:1500,])
p1 = plot_HP('Loading', melted_Loading)
p2 = plot_HP('Loading 1001-1500', melted_Loading_zoom)
gridExtra::grid.arrange(p1,p2, ncol = 2)</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-6-1.png" width="960" style="display: block; margin: auto;" /></p>
<p>The Loadings for the last 500 samples (1001-1500) have opposite sign for factor 1 and 2.</p>
<p>We perform flash analysis on L:</p>
<pre class="r"><code>f1.l = flash(Loadings1, greedy = TRUE, backfit = TRUE)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code>fitting factor/loading 4</code></pre>
<pre class="r"><code>f1l.f = flash_get_ldf(f1.l)$f
row.names(f1l.f) = paste0('Factor ', 1:3)
par(mfrow=c(1,3))
for(i in 1:ncol(f1l.f)){
barplot(f1l.f[,i], main = paste0('Factor ', i, ', pve=', round(flash_get_pve(f1.l)[i],3)))
}</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>We see that one loading is positive and the other is negative.</p>
<p>Check loadings (O) of Loadings:</p>
<pre class="r"><code>melted_LLoading = melt(flash_get_ldf(f1.l)$l * flash_get_ldf(f1.l)$d)
p3 = plot_HP('Loadings of Loadings', melted_LLoading)
p3</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-8-1.png" width="576" style="display: block; margin: auto;" /></p>
</div>
<div id="flash-with-fixed-factors-loading-same-sign" class="section level2">
<h2>Flash with fixed factors (loading same sign)</h2>
<pre class="r"><code>f2 = flash_add_fixed_f(data$Y, FF=cbind(c(1,1,1,1,1), c(1,1,0,0,0), c(0,0,0,0,1)))
f2 = flash_backfit(data$Y, f2)
Loading.fix = flash_get_ldf(f2)$l * flash_get_ldf(f2)$d
# Check loadings
melted_LoadingFix = melt(Loading.fix)
melted_LoadingFix_zoom = melt(Loading.fix[1001:1500,])
p4 = plot_HP('Loading', melted_LoadingFix)
p5 = plot_HP('Loading_zoom', melted_LoadingFix_zoom)
gridExtra::grid.arrange(p4,p5, ncol = 2)</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-9-1.png" width="960" style="display: block; margin: auto;" /></p>
<p>The Loadings for the last 500 samples (1001-1500) have same sign for factor 1 and 2.</p>
<p>Flash on loadings</p>
<pre class="r"><code>f2.l = flash(Loading.fix, greedy = TRUE, backfit = TRUE)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code>fitting factor/loading 4</code></pre>
<pre class="r"><code>f2l.f = flash_get_ldf(f2.l)$f
row.names(f2l.f) = paste0('Factor ', 1:3)
par(mfrow=c(1,3))
for(i in 1:ncol(f2l.f)){
barplot(f2l.f[,i], main = paste0('Factor ', i, ', pve=', round(flash_get_pve(f2.l)[i],3)))
}</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>Check Loading of Loaing</p>
<pre class="r"><code>melted_LLoading.fix = melt(flash_get_ldf(f2.l)$l * flash_get_ldf(f2.l)$d)
p6 = plot_HP('Loadings of Loadings', melted_LLoading.fix)
p6</code></pre>
<p><img src="figure/HierarchicalFlash.Rmd/unnamed-chunk-11-1.png" width="480" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="session-information" class="section level1">
<h1>Session information</h1>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.4 (2018-03-15)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.5
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] reshape2_1.4.3 ggplot2_3.0.0 flashr_0.5-12
loaded via a namespace (and not attached):
[1] Rcpp_0.12.17 bindr_0.1.1 compiler_3.4.4
[4] pillar_1.2.2 git2r_0.21.0 plyr_1.8.4
[7] iterators_1.0.9 tools_3.4.4 digest_0.6.15
[10] evaluate_0.10.1 tibble_1.4.2 gtable_0.2.0
[13] lattice_0.20-35 pkgconfig_2.0.1 rlang_0.2.1
[16] Matrix_1.2-14 foreach_1.4.4 yaml_2.1.19
[19] parallel_3.4.4 ebnm_0.1-12 bindrcpp_0.2.2
[22] gridExtra_2.3 withr_2.1.2 stringr_1.3.0
[25] dplyr_0.7.4 knitr_1.20 rprojroot_1.3-2
[28] grid_3.4.4 glue_1.2.0 R6_2.2.2
[31] rmarkdown_1.9 ashr_2.2-7 magrittr_1.5
[34] backports_1.1.2 scales_0.5.0 codetools_0.2-15
[37] htmltools_0.3.6 MASS_7.3-50 assertthat_0.2.0
[40] softImpute_1.4 colorspace_1.3-2 labeling_0.3
[43] stringi_1.2.2 lazyeval_0.2.1 pscl_1.5.2
[46] doParallel_1.0.11 munsell_0.4.3 truncnorm_1.0-8
[49] SQUAREM_2017.10-1</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>