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<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>
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<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>
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<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS Sierra 10.12.3
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] backports_1.0.5 magrittr_1.5 rprojroot_1.2 tools_3.3.2
[5] htmltools_0.3.5 yaml_2.1.14 Rcpp_0.12.10 stringi_1.1.2
[9] rmarkdown_1.3 knitr_1.15.1 stringr_1.1.0 digest_0.6.9
[13] evaluate_0.10 </code></pre>
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