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<h1 class="title toc-ignore">Fitting <span class="math inline">\(N\left(0, \sigma^2\right)\)</span> with Gaussian Derivatives</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-05-16</em></h4>
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<p><strong>Last updated:</strong> 2018-05-15</p>
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<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>We know the empirical distribution of a number of correlated null <span class="math inline">\(N\left(0, 1\right)\)</span> <span class="math inline">\(z\)</span> scores can be approximated by Gaussian derivatives at least theoretically. Now we want to know if the <span class="math inline">\(N\left(0, \sigma^2\right)\)</span> density can also be approximated by Gaussian derivatives. The question is important because it is related to the identifiability between the correlated null and true signals using the tool of Gaussian derivatives.</p>
<p>The same question was investigated for <a href="alternative.html">small <span class="math inline">\(\sigma^2\)</span></a> and <a href="alternative2.html">large <span class="math inline">\(\sigma^2\)</span></a>. But these investigations was <em>empirical</em>, in the sense that we are fitting the Gaussian derivatives to a large number of independent samples simulated from <span class="math inline">\(N\left(0, \sigma^2\right)\)</span>. Now we are doing that <strong>theoretically</strong>.</p>
</div>
<div id="hermite-moments" class="section level2">
<h2>Hermite moments</h2>
<p>If a probability density function (pdf) <span class="math inline">\(f\left(x\right)\)</span> can be approximated by normalized Gaussian derivatives, we should have</p>
<p><span class="math display">\[
f\left(x\right) = w_0\varphi\left(x\right) + w_1\varphi^{(1)}\left(x\right) + w_2\frac{1}{\sqrt{2!}}\varphi^{(2)}\left(x\right) + \cdots + w_n\frac{1}{\sqrt{n!}}\varphi^{(n)}\left(x\right) + \cdots \ .
\]</span> Using the fact that Hermite polynomials are orthonormal with respect to the Gaussian kernel and normalizing constants, we have</p>
<p><span class="math display">\[
\int_{\mathbb{R}}
\frac{1}{\sqrt{m!}}
h_m\left(x\right) \frac{1}{\sqrt{n!}}\varphi^{(n)}\left(x\right)dx = \left(-1\right)^n\delta_{mn} \ .
\]</span> Therefore, when using Gaussian derivatives to decompose a pdf <span class="math inline">\(f\)</span>, the coefficients <span class="math inline">\(w_n\)</span> can be obtained by</p>
<p><span class="math display">\[
w_n = \left(-1\right)^n \int \frac{1}{\sqrt{n!}} h_n(x) f(x) dx = \frac{\left(-1\right)^n}{\sqrt{n!}} E_f\left[h_n\left(x\right)\right] \ .
\]</span> Since <span class="math inline">\(h_n\)</span> is a degree <span class="math inline">\(n\)</span> polynomials, <span class="math inline">\(E_f\left[h_n\left(x\right)\right]\)</span> is essentially a linear combination of relevant moments up to order <span class="math inline">\(n\)</span>, denoted as <span class="math inline">\(E\left[x^i\right]\)</span>. For example, here are the first several <span class="math inline">\(E_f\left[h_n\left(x\right)\right]\)</span>.</p>
<p><span class="math display">\[
\begin{array}{rclcl}
E_f\left[h_0\left(x\right)\right] &=& E_f\left[1\right] &=& 1 \\
E_f\left[h_1\left(x\right)\right] &=& E_f\left[x\right] &=& E\left[x\right] \\
E_f\left[h_2\left(x\right)\right] &=& E_f\left[x^2 - 1\right] &=& E\left[x^2\right] - 1\\
E_f\left[h_3\left(x\right)\right] &=& E_f\left[x^3 - 3x\right] &=& E\left[x^3\right] - 3 E\left[x\right]\\
E_f\left[h_4\left(x\right)\right] &=& E_f\left[x^4 - 6x^2 + 3\right] &=& E\left[x^4\right] - 6 E\left[x^2\right] + 3\\
E_f\left[h_5\left(x\right)\right] &=& E_f\left[x^5 - 10x^3 + 15x\right] &=& E\left[x^5\right] - 10 E\left[x^3\right] + 15 E\left[x\right]\\
E_f\left[h_6\left(x\right)\right] &=& E_f\left[x^6 - 15x^4 + 45x^2 - 15\right] &=& E\left[x^6\right] - 15 E\left[x^4\right] + 45E\left[x^2\right] - 15\\
\end{array}
\]</span> Some call these “Hermite moments.”</p>
</div>
<div id="when-f-nleft0-sigma2right-general-results" class="section level2">
<h2>When <span class="math inline">\(f = N\left(0, \sigma^2\right)\)</span>: General results</h2>
<p>When <span class="math inline">\(f = N\left(0, \sigma^2\right)\)</span>, we can write all Hermite moments out analytically. After algebra, it turns out they can be written as</p>
<p><span class="math display">\[
E_f\left[h_n\left(x\right)\right] =
\begin{cases}
0 & n \text{ is odd}\\
\left(n - 1\right)!! \left(\sigma^2 - 1\right)^{n / 2} & n \text{ is even}
\end{cases} \ ,
\]</span> where <span class="math inline">\(\left(n - 1\right)!!\)</span> is the <a href="https://en.wikipedia.org/wiki/Double_factorial">double factorial</a>.</p>
<p>Hence, if we want to express the pdf of <span class="math inline">\(N\left(0, \sigma^2\right)\)</span> in terms of normalized Gaussian derivatives, the coefficient</p>
<p><span class="math display">\[
w_n =
\begin{cases}
0 & n \text{ is odd}\\
\frac{\left(n - 1\right)!!}{\sqrt{n!}}\left(\sigma^2 - 1\right)^{n / 2}
=\sqrt{\frac{\left(n - 1\right)!!}{n!!}}\left(\sigma^2 - 1\right)^{n / 2}
& n \text{ is even}
\end{cases} \ .
\]</span> From now on we will only pay attention to the even-order coefficients <span class="math inline">\(w_n = \sqrt{\frac{\left(n - 1\right)!!}{n!!}}\left(\sigma^2 - 1\right)^{n / 2} := U_n\left(\sigma^2 - 1\right)^{n / 2}\)</span>, where the sequence <span class="math inline">\(U_n := \sqrt{\frac{\left(n - 1\right)!!}{n!!}}\)</span>. Note that this sequence <span class="math inline">\(U_n\)</span> is interesting. <a href="https://math.stackexchange.com/questions/584456/limit-of-a-fraction-of-double-factorials">It can be proved</a> that it’s going to zero as <span class="math inline">\(n\to\infty\)</span>. Actually it can be <a href="https://en.wikipedia.org/wiki/Double_factorial#Additional_identities">written in another way</a></p>
<p><span class="math display">\[
\int_{0}^{\frac\pi2} \sin^nx dx = \int_{0}^{\frac\pi2} \cos^nx dx = \frac\pi2U_n^2 \ .
\]</span> However, this sequence decays slowly. In particular, it doesn’t decay exponentially, as seen in the following plots.</p>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
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<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/fitting_normal.rmd/unnamed-chunk-3-1.png" target="_blank">0f36d99</a>
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<a href="https://github.com/LSun/truncash/blob/a5300252ff521211d8a6c6a80a7cbb58b4d89438/docs/figure/fitting_normal.rmd/unnamed-chunk-3-1.png" target="_blank">a530025</a>
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<p><img src="figure/fitting_normal.rmd/unnamed-chunk-3-2.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/fitting_normal.rmd/unnamed-chunk-3-2.png" target="_blank">0f36d99</a>
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<p><strong>Therefore, when <span class="math inline">\(\sigma^2 > 2\)</span>, <span class="math inline">\(w_n\)</span> is exploding!</strong> It means theoretially we cannot actually fit <span class="math inline">\(N\left(0, \sigma^2\right)\)</span> with Gaussian derivatives when <span class="math inline">\(\sigma^2 > 2\)</span>.</p>
<p>It has multiple implications. First, it shows why people usually say Gaussian derivatives can only fit a density that’s close enough to Gaussian, especially, a density whose variance is too “inflated” compared with the standard normal. Therefore, the fact that in many case, the correlated null can be satisfactorily fitted by Gaussian derivatives tells us the inflation caused by correlation is indeed peculiar.</p>
<p>On the other hand, when <span class="math inline">\(0 < \sigma^2 < 2\)</span>, that is, <span class="math inline">\(\left|\sigma^2 - 1\right| < 1\)</span>, <span class="math inline">\(w_n\)</span> decays exponentially. It means that we are able to satisfactorily fit <span class="math inline">\(N\left(0, \sigma^2\right)\)</span> with a limited number of Gaussian derivatives such that</p>
<p><span class="math display">\[
f_N\left(x\right) = \varphi\left(x\right) + \sum\limits_{n = 1}^N w_n\frac{1}{\sqrt{n!}}\varphi^{(n)}\left(x\right) \approx N\left(0, \sigma^2\right) = \frac{1}{\sigma}\varphi\left(\frac{x}{\sigma}\right) \ .
\]</span></p>
<p>It also means that when the correlated null looks like <span class="math inline">\(N\left(0, \sigma^2\right)\)</span> with <span class="math inline">\(\sigma^2\in\left(1, 2\right]\)</span> (small inflation) or <span class="math inline">\(\sigma^2\in\left(0, 1\right)\)</span> (deflation), it’s hard to identify whether the inflation or deflation is caused by correlation or true effects.</p>
</div>
<div id="when-f-nleft0-sqrt22right-in-particular" class="section level2">
<h2>When <span class="math inline">\(f = N\left(0, \sqrt{2}^2\right)\)</span> in particular</h2>
<p>That’s why in previous empirical investigations, we found that we <a href="alternative.html">could fit <span class="math inline">\(N\left(0, \sqrt{2}^2\right)\)</span> relatively well</a>, but <a href="alternative2.html">could not fit when <span class="math inline">\(\sigma^2 > 2\)</span></a>.</p>
<p>Actually, <strong><span class="math inline">\(N\left(0, \sqrt{2}^2\right)\)</span> is a singularly interesting case</strong>. This case corresponds to when we have <span class="math inline">\(N\left(0, 1\right)\)</span> signal and <span class="math inline">\(N\left(0, 1\right)\)</span> independent noise, hence <span class="math inline">\(SNR = 0\)</span>. Theoretically the density curve can be fitted when the odd-order coefficients are <span class="math inline">\(0\)</span> and the even-order ones are <span class="math inline">\(U_n = \sqrt{\frac{\left(n - 1\right)!!}{n!!}}\)</span>.</p>
<p>However, since <span class="math inline">\(U_n\)</span> is not decaying fast enough, the re-constructed curve using a limited number of Gaussian derivatives doesn’t look good. Meanwhile, if we truly have many random samples from <span class="math inline">\(N\left(0, \sqrt{2}^2\right)\)</span>, we can fit the data using Gaussian derivatives, and the estimated coefficients <span class="math inline">\(\hat w\)</span> usually give a better fit, as <a href="alternative.html">seen previously</a>.</p>
<p>Here we plot the fitted curve using first <span class="math inline">\(N = 10, 50, 100\)</span> orders of Gaussian derivatives, compared with the true <span class="math inline">\(N\left(0, \sqrt{2}^2\right)\)</span> density and the curve estimated from random samples using <span class="math inline">\(L = 10\)</span> Gaussian derivatives.</p>
<pre><code>Estimated Normalized w: 0 ~ 1 ; 1 ~ -0.0126 ; 2 ~ 0.72013 ; 3 ~ -0.01709 ; 4 ~ 0.6782 ; 5 ~ 0.01241 ; 6 ~ 0.62481 ; 7 ~ 0.03822 ; 8 ~ 0.4004 ; 9 ~ 0.00207 ; 10 ~ 0.11385 ; </code></pre>
<pre><code>Theoretical Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ 0.70711 ; 3 ~ 0 ; 4 ~ 0.61237 ; 5 ~ 0 ; 6 ~ 0.55902 ; 7 ~ 0 ; 8 ~ 0.52291 ; 9 ~ 0 ; 10 ~ 0.49608 ; ...</code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
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<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/fitting_normal.rmd/unnamed-chunk-4-1.png" target="_blank">0f36d99</a>
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</div>
<div id="sigma2-2-multiple-examples" class="section level2">
<h2><span class="math inline">\(0 < \sigma^2 < 2\)</span>: Multiple examples</h2>
<div id="sigma2-0.2-n-40" class="section level3">
<h3><span class="math inline">\(\sigma^2 = 0.2\)</span>, <span class="math inline">\(N = 40\)</span></h3>
<pre><code>Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ -0.56569 ; 3 ~ 0 ; 4 ~ 0.39192 ; 5 ~ 0 ; 6 ~ -0.28622 ; 7 ~ 0 ; 8 ~ 0.21418 ; 9 ~ 0 ; 10 ~ -0.16255 ; 11 ~ 0 ; 12 ~ 0.12451 ; 13 ~ 0 ; 14 ~ -0.09598 ; 15 ~ 0 ; 16 ~ 0.07435 ; 17 ~ 0 ; 18 ~ -0.0578 ; 19 ~ 0 ; 20 ~ 0.04507 ; 21 ~ 0 ; 22 ~ -0.03523 ; 23 ~ 0 ; 24 ~ 0.02759 ; 25 ~ 0 ; 26 ~ -0.02164 ; 27 ~ 0 ; 28 ~ 0.017 ; 29 ~ 0 ; 30 ~ -0.01337 ; 31 ~ 0 ; 32 ~ 0.01053 ; 33 ~ 0 ; 34 ~ -0.0083 ; 35 ~ 0 ; 36 ~ 0.00655 ; 37 ~ 0 ; 38 ~ -0.00517 ; 39 ~ 0 ; 40 ~ 0.00408 ; </code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="sigma2-0.5-n-14" class="section level3">
<h3><span class="math inline">\(\sigma^2 = 0.5\)</span>, <span class="math inline">\(N = 14\)</span></h3>
<pre><code>Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ -0.35355 ; 3 ~ 0 ; 4 ~ 0.15309 ; 5 ~ 0 ; 6 ~ -0.06988 ; 7 ~ 0 ; 8 ~ 0.03268 ; 9 ~ 0 ; 10 ~ -0.0155 ; 11 ~ 0 ; 12 ~ 0.00742 ; 13 ~ 0 ; 14 ~ -0.00358 ; </code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="sigma2-0.8-n-6" class="section level3">
<h3><span class="math inline">\(\sigma^2 = 0.8\)</span>, <span class="math inline">\(N = 6\)</span></h3>
<pre><code>Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ -0.14142 ; 3 ~ 0 ; 4 ~ 0.02449 ; 5 ~ 0 ; 6 ~ -0.00447 ; </code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="sigma2-1.2-n-6" class="section level3">
<h3><span class="math inline">\(\sigma^2 = 1.2\)</span>, <span class="math inline">\(N = 6\)</span></h3>
<pre><code>Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ 0.14142 ; 3 ~ 0 ; 4 ~ 0.02449 ; 5 ~ 0 ; 6 ~ 0.00447 ; </code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="sigma2-1.5-n-14" class="section level3">
<h3><span class="math inline">\(\sigma^2 = 1.5\)</span>, <span class="math inline">\(N = 14\)</span></h3>
<pre><code>Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ 0.35355 ; 3 ~ 0 ; 4 ~ 0.15309 ; 5 ~ 0 ; 6 ~ 0.06988 ; 7 ~ 0 ; 8 ~ 0.03268 ; 9 ~ 0 ; 10 ~ 0.0155 ; 11 ~ 0 ; 12 ~ 0.00742 ; 13 ~ 0 ; 14 ~ 0.00358 ; </code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-9-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="sigma2-1.8-n-40" class="section level3">
<h3><span class="math inline">\(\sigma^2 = 1.8\)</span>, <span class="math inline">\(N = 40\)</span></h3>
<pre><code>Normalized w: 0 ~ 1 ; 1 ~ 0 ; 2 ~ 0.56569 ; 3 ~ 0 ; 4 ~ 0.39192 ; 5 ~ 0 ; 6 ~ 0.28622 ; 7 ~ 0 ; 8 ~ 0.21418 ; 9 ~ 0 ; 10 ~ 0.16255 ; 11 ~ 0 ; 12 ~ 0.12451 ; 13 ~ 0 ; 14 ~ 0.09598 ; 15 ~ 0 ; 16 ~ 0.07435 ; 17 ~ 0 ; 18 ~ 0.0578 ; 19 ~ 0 ; 20 ~ 0.04507 ; 21 ~ 0 ; 22 ~ 0.03523 ; 23 ~ 0 ; 24 ~ 0.02759 ; 25 ~ 0 ; 26 ~ 0.02164 ; 27 ~ 0 ; 28 ~ 0.017 ; 29 ~ 0 ; 30 ~ 0.01337 ; 31 ~ 0 ; 32 ~ 0.01053 ; 33 ~ 0 ; 34 ~ 0.0083 ; 35 ~ 0 ; 36 ~ 0.00655 ; 37 ~ 0 ; 38 ~ 0.00517 ; 39 ~ 0 ; 40 ~ 0.00408 ; </code></pre>
<p><img src="figure/fitting_normal.rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="another-special-case-sigma2-0" class="section level2">
<h2>Another special case: <span class="math inline">\(\sigma^2 = 0\)</span></h2>
<p>When <span class="math inline">\(\sigma^2 = 0\)</span>, <span class="math inline">\(f = N\left(0, 0\right) = \delta_0\)</span>. In this case,</p>
<p><span class="math display">\[
w_n =
\begin{cases}
0 & n \text{ is odd}\\
\sqrt{\frac{\left(n - 1\right)!!}{n!!}}\left(-1\right)^{n / 2}
& n \text{ is even}
\end{cases}
\]</span> is equivalent to what we’ve <a href="gaussian_derivatives_4.html#extreme_case:_(rho_%7Bij%7D_equiv_1)">obtained previously</a></p>
<p><span class="math display">\[
w_n = \frac{1}{\sqrt{n!}}h_n\left(0\right) \ .
\]</span> The performance can be seen <a href="gd_delta.html#(z_=_0)">here</a>.</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] Rmosek_8.0.69 CVXR_0.95 REBayes_1.2 Matrix_1.2-12
[5] SQUAREM_2017.10-1 EQL_1.0-0 ttutils_1.0-1 PolynomF_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 ECOSolveR_0.4
[22] R.utils_2.6.0 evaluate_0.10.1 rmarkdown_1.9
[25] stringi_1.1.6 Rmpfr_0.6-1 compiler_3.4.3
[28] backports_1.1.2 R.methodsS3_1.7.1</code></pre>
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