<!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: Large Correlation</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: Large Correlation</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_4.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_4.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_4.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/66399687da141ce91db2cd3da67f243c80436f67/docs/gaussian_derivatives_4.html" target="_blank">6639968</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-11-05
</td>
<td style="text-align:left;">
transfer
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7344c2d6c85eeaa4a51e948f9177261ba4c502b5/analysis/gaussian_derivatives_4.rmd" target="_blank">7344c2d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-03-30
</td>
<td style="text-align:left;">
constraints
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/7344c2d6c85eeaa4a51e948f9177261ba4c502b5/docs/gaussian_derivatives_4.html" target="_blank">7344c2d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-03-30
</td>
<td style="text-align:left;">
constraints
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/5bd5d70c2467c0ebba78974f24b8603e18800886/analysis/gaussian_derivatives_4.rmd" target="_blank">5bd5d70</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-03-28
</td>
<td style="text-align:left;">
revision
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/5bd5d70c2467c0ebba78974f24b8603e18800886/docs/gaussian_derivatives_4.html" target="_blank">5bd5d70</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-03-28
</td>
<td style="text-align:left;">
revision
</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_4.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_4.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="can-k-be-too-small" class="section level2">
<h2>Can <span class="math inline">\(K\)</span> be too small?</h2>
<p><a href="gaussian_derivatives.rmd">Another assumption</a> to make the problem tractable is that the pairwise correlation <span class="math inline">\(\rho_{ij}\)</span> is moderate enough so <span class="math inline">\(W_k\varphi^{(k)}\)</span> vanishes as the order <span class="math inline">\(k\)</span> increases. With this assumption we can stop at a sufficiently large <span class="math inline">\(K\)</span> without consideration higher order Gaussian derivatives. But what if <span class="math inline">\(\rho_{ij}\)</span> is large?</p>
</div>
<div id="extreme-case-rho_ij-equiv-1" class="section level2">
<h2>Extreme case: <span class="math inline">\(\rho_{ij} \equiv 1\)</span></h2>
<p>When we have perfect correlation among all <span class="math inline">\(z\)</span> scores, the approximate limit observed density <span class="math inline">\(f_0(x)\to\delta_z(x) = \delta(x-z)\)</span>. That is, with probability one, we observe <span class="math inline">\(z_1 = \cdots = z_n = z\)</span>, as <span class="math inline">\(n\to\infty\)</span>, <span class="math inline">\(f_0(x)\)</span> goes to a Dirac delta function peak at the observed <span class="math inline">\(z\)</span>, and zero elsewhere. Now the question is, can this Dirac delta function be decomposed with the Gaussian <span class="math inline">\(\varphi\)</span> and its derivatives <span class="math inline">\(\varphi^{(k)}\)</span>, so that we still have</p>
<p><span class="math display">\[
f_0(x) = \delta(x - z) = \varphi(x)\sum\limits_{k = 0}^\infty W_kh_k(x)
\]</span> with appropriate <span class="math inline">\(W_k\)</span>’s?</p>
<p>Using the orthogonality of Hermite functions, <a href="gaussian_derivatives.html#gaussian_derivatives_and_hermite_polynomials">we have</a></p>
<p><span class="math display">\[
W_k = \frac{1}{k!}\int_{-\infty}^{\infty}h_k(x)f_0(x)dx = \frac{1}{k!}\int_{-\infty}^{\infty}h_k(x)\delta(x-z)dx = \frac{1}{k!}h_k(z)
\]</span> Now the decomposition <span class="math inline">\(\varphi(x)\sum\limits_{k = 0}^\infty W_kh_k(x)\)</span> becomes</p>
<p><span class="math display">\[
\varphi(x)\sum\limits_{k = 0}^\infty \frac{1}{k!}h_k(z)h_k(x)
\]</span> It turns out this equation is connected to <a href="https://en.wikipedia.org/wiki/Mehler_kernel">Mehler’s formula</a> which can be <a href="https://en.wikipedia.org/wiki/Hermite_polynomials#Completeness_relation">shown</a> to give the identity</p>
<p><span class="math display">\[
\sum\limits_{k = 0}^\infty \psi_k(x)\psi_k(z) = \delta(x - z)
\]</span> where <span class="math inline">\(\psi_k\)</span>’s are the <a href="https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions">Hermite functions</a> defined as</p>
<p><span class="math display">\[
\begin{array}{rrcl}
& \psi_k(x) &=& (k!)^{-1/2}(\sqrt{\pi})^{-1/2}e^{-x^2/2}h_k(\sqrt{2}x)\\
\Rightarrow & h_k(x) &=& (k!)^{1/2}(\sqrt{\pi})^{1/2}e^{x^2/4}\psi_k\left(\frac x{\sqrt{2}}\right)\\
\Rightarrow & \varphi(x)\sum\limits_{k = 0}^\infty \frac{1}{k!}h_k(z)h_k(x) & =&
\frac1{\sqrt{2}}e^{-\frac{x^2}4+\frac{z^2}4}\sum\limits_{k = 0}^\infty
\psi_k\left(\frac x{\sqrt{2}}\right)\psi_k\left(\frac z{\sqrt{2}}\right)\\
& &=&
\frac1{\sqrt{2}}e^{-\frac{x^2}4+\frac{z^2}4}
\delta\left(\frac{x - z}{\sqrt{2}}\right)
\end{array}
\]</span> Note that the Dirac delta function has a property that <span class="math inline">\(\delta(\alpha x) = \delta(x) / |\alpha| \Rightarrow \frac1{\sqrt{2}}\delta\left(\frac{x - z}{\sqrt{2}}\right) = \delta(x - z)\)</span>. Therefore,</p>
<p><span class="math display">\[
\varphi(x)\sum\limits_{k = 0}^\infty \frac{1}{k!}h_k(z)h_k(x)
=
\delta(x-z)\exp\left(-\frac{x^2}4+\frac{z^2}4\right)
\]</span> Note that <span class="math inline">\(\exp\left(-\frac{x^2}4+\frac{z^2}4\right)\)</span> is bounded for any <span class="math inline">\(z\in\mathbb{R}\)</span>, so <span class="math inline">\(\delta(x-z)\exp\left(-\frac{x^2}4+\frac{z^2}4\right)\)</span> vanishes to <span class="math inline">\(0\)</span> for any <span class="math inline">\(x\neq z\)</span>, and</p>
<p><span class="math display">\[
\int_{-\infty}^\infty \delta(x-z)\exp\left(-\frac{x^2}4+\frac{z^2}4\right)dx =
\exp\left(-\frac{z^2}4+\frac{z^2}4\right) = 1
\]</span> Hence, in essence, <span class="math inline">\(\delta(x-z)\exp\left(-\frac{x^2}4+\frac{z^2}4\right) = \delta(x-z)\)</span>. Therefore we have <span class="math display">\[
f_0(x) = \varphi(x)\sum\limits_{k = 0}^\infty W_kh_k(x)
=
\varphi(x)\sum\limits_{k = 0}^\infty \frac{1}{k!}h_k(z)h_k(x)
=\delta(x -z)
\]</span> when <span class="math display">\[
W_k = \frac{1}{k!}h_k(z)
\]</span> Thus we show that the Dirac delta function can be decomposed by Gaussian density and its derivatives.</p>
<div id="visualization-with-finite-k" class="section level3">
<h3>Visualization with finite <span class="math inline">\(K\)</span></h3>
<p>With Gaussian and its infinite orders of derivatives, we can compose a Dirac delta function at any position, yet what happens if we stop at a finite <span class="math inline">\(K\)</span>? Let <span class="math inline">\(f_0^K\)</span> be the approximation of <span class="math inline">\(f_0 = \delta_z\)</span> with first <span class="math inline">\(K\)</span> Gaussian derivatives. That is,</p>
<p><span class="math display">\[
f_0^K(x) = \varphi(x)\sum\limits_{k = 0}^K \frac{1}{k!}h_k(z)h_k(x) \ .
\]</span> Meanwhile, let <span class="math inline">\(F_0^K(x) = \int_{-\infty}^x f_0^K(u)du\)</span>. It’s easy to shown that</p>
<p><span class="math display">\[
F_0^K(x) = \Phi(x) - \varphi(x)\sum\limits_{k = 1}^K W_k h_{k - 1}(x) = \Phi(x) - \varphi(x) \sum\limits_{k = 1}^K \frac{1}{k!}h_k(z) h_{k - 1}(x) \ .
\]</span></p>
<p>Theoretically, <span class="math inline">\(f_0^K\)</span> is an approximation to empirical density of perfectly correlated <span class="math inline">\(z\)</span> scores; hence, as <span class="math inline">\(K\to\infty\)</span>, <span class="math inline">\(f_0^K\to\delta_z\)</span>. Similarly, <span class="math inline">\(F_0^K\)</span> is an approximation to empirical cdf of perfectly correlated <span class="math inline">\(z\)</span> scores; hence, as <span class="math inline">\(K\to\infty\)</span>, <span class="math inline">\(f_0^K\)</span> should converge to the <span class="math inline">\(0\)</span>-<span class="math inline">\(1\)</span> step function, and the location of the step is the observed <span class="math inline">\(z\)</span>.</p>
<p>In practice, the convergence is not fast. As we can see from the following visualization, the difference between <span class="math inline">\(f_0^K\)</span> and <span class="math inline">\(\delta_z\)</span>, as well as that between <span class="math inline">\(F_0^K\)</span> and the step function, is still conspicuous even if <span class="math inline">\(K = 20\)</span>, which is about the highest order <code>R</code> can reasonbly handle in the current implementation. Therefore, at least in theory it’s possible that <span class="math inline">\(K\)</span> can be too small.</p>
<p><strong>Note that the oscillation near the presumptive step may be connected with <a href="https://en.wikipedia.org/wiki/Gibbs_phenomenon">Gibbs phenomenon</a>.</strong></p>
<pre><code>Under perfect correlation, observed z scores = -1 </code></pre>
<p><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-2-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-2-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Under perfect correlation, observed z scores = 0 </code></pre>
<p><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-2-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-2-4.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Under perfect correlation, observed z scores = 2 </code></pre>
<p><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-2-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-2-6.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="fitting-experiments-when-rho_ij-is-large" class="section level2">
<h2>Fitting experiments when <span class="math inline">\(\rho_{ij}\)</span> is large</h2>
<p>As previous theoretical result indicates, when <span class="math inline">\(\rho\)</span> is large, a large <span class="math inline">\(K\)</span> is probably needed. However, on the other hand, when <span class="math inline">\(\rho\)</span> is large, the effective sample size is small. Indeed when <span class="math inline">\(\rho\equiv1\)</span>, the sample size is essentially <span class="math inline">\(1\)</span>.</p>
<p>Let’s take a look at some examples with pairwise correlations of <span class="math inline">\(z\)</span> scores <span class="math inline">\(\rho_{ij}\equiv\rho\)</span>, <span class="math inline">\(\rho\)</span> moderate to high. Such <span class="math inline">\(z\)</span> scores can be simulated as <span class="math inline">\(z_i = \epsilon\sqrt{\rho} + e_i\sqrt{1-\rho}\)</span>, where <span class="math inline">\(\epsilon, e_i\)</span> are iid <span class="math inline">\(N(0, 1)\)</span>.</p>
<pre class="r"><code>n = 1e4
rho = 0.5
set.seed(777)
z = rnorm(1) * sqrt(rho) + rnorm(n) * sqrt(1 - rho)</code></pre>
<pre class="r"><code>source("../code/ecdfz.R")
fit.ecdfz = ecdfz.optimal(z)</code></pre>
<p>When <span class="math inline">\(\rho = 0.5\)</span>, current implementation with <span class="math inline">\(K = 5\)</span> fits positively correlationed z scores reasonably well.</p>
<pre><code>10000 z scores with pairwise correlation = 0.5</code></pre>
<p><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>However, as <span class="math inline">\(\rho\)</span> gets larger, current implementation usually fails to find a good <span class="math inline">\(K\)</span> before the algorithm goes unstable, as illustrated in the following <span class="math inline">\(\rho = 0.7\)</span> plot. <span class="math inline">\(K = 3\)</span> is obviously not enough, yet <span class="math inline">\(K = 4\)</span> has already gone wildly unstable.</p>
<pre class="r"><code>n = 1e4
rho = 0.7
set.seed(777)
z = rnorm(1) * sqrt(rho) + rnorm(n) * sqrt(1 - rho)</code></pre>
<pre class="r"><code>source("../code/ecdfz.R")
fit.ecdfz = ecdfz.optimal(z)</code></pre>
<pre><code>10000 z scores with pairwise correlation = 0.7</code></pre>
<p><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-9-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>When <span class="math inline">\(\rho = 0.9\)</span>, the observed <span class="math inline">\(z\)</span> scores are so concentrated in a small range, even if we have <span class="math inline">\(10,000\)</span> of them, making the effective sample size hopelessly small. Current implementation can’t even handle this data set; it goes crazy when <span class="math inline">\(K = 2\)</span>.</p>
<pre class="r"><code>n = 1e4
rho = 0.9
set.seed(777)
z = rnorm(1) * sqrt(rho) + rnorm(n) * sqrt(1 - rho)</code></pre>
<pre class="r"><code>source("../code/ecdfz.R")
fit.ecdfz = ecdfz(z, 2)</code></pre>
<pre><code>10000 z scores with pairwise correlation = 0.9</code></pre>
<p><img src="figure/gaussian_derivatives_4.rmd/unnamed-chunk-12-1.png" width="672" style="display: block; margin: auto;" /></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
loaded via a namespace (and not attached):
[1] workflowr_1.0.1 Rcpp_0.12.16 digest_0.6.15
[4] rprojroot_1.3-2 R.methodsS3_1.7.1 backports_1.1.2
[7] git2r_0.21.0 magrittr_1.5 evaluate_0.10.1
[10] stringi_1.1.6 whisker_0.3-2 R.oo_1.21.0
[13] R.utils_2.6.0 rmarkdown_1.9 tools_3.4.3
[16] stringr_1.3.0 yaml_2.1.18 compiler_3.4.3
[19] htmltools_0.3.6 knitr_1.20 </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>