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<title>True Signal vs Correlated Null: Identifiability &amp; Small Effects</title>

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<h1 class="title toc-ignore">True Signal vs Correlated Null: Identifiability &amp; Small Effects</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-03-29</em></h4>

</div>


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<p><strong>Last updated:</strong> 2017-05-30</p>
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<p><strong>Code version:</strong> 9f7dd01</p>
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<div id="identifiablity-of-true-signals-from-correlated-noise" class="section level2">
<h2>Identifiablity of true signals from correlated noise</h2>
<p><a href="gaussian_derivatives_2.html#examples">We’ve shown</a> that in many real data sets when we have correlated null <span class="math inline">\(z\)</span> scores, we can <a href="gaussian_derivatives.html#empirical_null">fit their empirical distribution with Gaussian and its derivatives</a>.</p>
<p>But what if we have true signals instead of the global null? Theoretically, any distribution can be decomposed by Gaussian and its derivatives, also called <a href="https://en.wikipedia.org/wiki/Edgeworth_series">Edgeworth series or Edgeworth expansion</a>. We’ve shown that the Dirac delta function <span class="math inline">\(\delta_z\)</span> and the associated <span class="math inline">\(0\)</span>-<span class="math inline">\(1\)</span> step function <a href="gaussian_derivatives_4.html#extreme_case:_(rho_%7Bij%7D_equiv_1)">can be decomposed</a> by Gaussian derivatives. Essentially all distributions can be represented by (usually infinitely many) <span class="math inline">\(\delta_z\)</span>, and thus be decomposed by Gaussian and its derivatives. <strong>There is a rich literature on this topic, probably of further use to this project.</strong></p>
<p>Now the more urgent problem is: can true signals also be fitted by Gaussian derivatives in a similar way as correlated null? Let normalized weights <span class="math inline">\(W_k^s = W_k\sqrt{k!}\)</span>. As <a href="gaussian_derivatives.html">shown previously</a>, under correlated null, the variance <span class="math inline">\(\text{var}(W_k^s) = \alpha_k = \bar{\rho_{ij}^k}\)</span>. Thus, under correlated null, the Gaussian derivative decomposition of the empirical distribution should have “reasonable” weights of similar decaying patterns.</p>
<p>If it turns out Gaussian derivatives with limited orders (say, <span class="math inline">\(K \leq 10\)</span>) and reasonable normalized weights are only able to fit the empirical correlated null, but nothing else, then properly regularized Gaussian derivatives can be readily used to control the usually correlated <em>noise</em>, which are correlated null, and leave the <em>signal</em> to <code>ash</code>. But if true signals can also be fitted this way, the identifiability of true signals from correlated noise becomes an issue.</p>
<p>Let’s start with the simplest case: <span class="math inline">\(z \sim N(0, \sqrt{2}^2)\)</span> independently. This data set can be seen as generated as follows.</p>
<p><span class="math display">\[
\begin{array}{c}
\beta_j \sim N(0, 1)\\
z_j \sim N(\beta_j, 1)
\end{array}
\]</span></p>
<p>That is, a <span class="math inline">\(N(0, 1)\)</span> true signal is polluted by a <span class="math inline">\(N(0, 1)\)</span> noise.</p>
</div>
<div id="illustration" class="section level2">
<h2>Illustration</h2>
<pre class="r"><code>n = 1e4
m = 5
set.seed(777)
zmat = matrix(rnorm(n * m, 0, sd = sqrt(2)), nrow = m, byrow = TRUE)</code></pre>
<pre class="r"><code>library(ashr)
source(&quot;../code/ecdfz.R&quot;)
res = list()
for (i in 1:m) {
  z = zmat[i, ]
  p = (1 - pnorm(abs(z))) * 2
  bh.fd = sum(p.adjust(p, method = &quot;BH&quot;) &lt;= 0.05)
  pihat0.ash = get_pi0(ash(z, 1, method = &quot;fdr&quot;))
  ecdfz.fit = ecdfz.optimal(z)
  res[[i]] = list(z = z, p = p, bh.fd = bh.fd, pihat0.ash = pihat0.ash, ecdfz.fit = ecdfz.fit)
}</code></pre>
<pre><code>Example 1 : Number of Discoveries: 246 ; pihat0 = 0.3245191 
Log-likelihood with N(0, 2): -17704.62 
Log-likelihood with Gaussian Derivatives: -17702.15 
Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: -2.473037 
Normalized weights:
1 : -0.0126888368547959 ; 2 : 0.717062378249889 ; 3 : -0.0184536200134752 ; 4 : 0.649465525394262 ; 5 : 0.00859163522314002 ; 6 : 0.521325079359314 ; 7 : 0.0334885164431775 ; 8 : 0.22636494735755 ;</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the left tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the right tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-8.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Example 2 : Number of Discoveries: 218 ; pihat0 = 0.3007316 
Log-likelihood with N(0, 2): -17620.91 
Log-likelihood with Gaussian Derivatives: -17618.13 
Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: -2.787631 
Normalized weights:
1 : 0.0102680011779709 ; 2 : 0.696012169853609 ; 3 : 0.0113000171720435 ; 4 : 0.544236663386519 ; 5 : -0.0208432030918437 ; 6 : 0.359654087688657 ; 7 : 0.00449356234470338 ; 8 : 0.129368209367989 ;</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-9.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the left tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-10.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the right tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-16.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Example 3 : Number of Discoveries: 201 ; pihat0 = 0.3524008 
Log-likelihood with N(0, 2): -17627.66 
Log-likelihood with Gaussian Derivatives: -17623.26 
Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: -4.397359 
Normalized weights:
1 : 0.000611199281683122 ; 2 : 0.697833563596919 ; 3 : -9.24232505276873e-05 ; 4 : 0.593310577011007 ; 5 : 0.0690423192366928 ; 6 : 0.402719962212205 ; 7 : 0.0821756084741036 ; 8 : 0.137136244590824 ;</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-17.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the left tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-18.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the right tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-21.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-22.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-23.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-24.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Example 4 : Number of Discoveries: 134 ; pihat0 = 0.3039997 
Log-likelihood with N(0, 2): -17572.28 
Log-likelihood with Gaussian Derivatives: -17589.35 
Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: 17.07424 
Normalized weights:
1 : -0.00303021567753385 ; 2 : 0.667140676046508 ; 3 : -0.00744442518950379 ; 4 : 0.4335954662891 ; 5 : 0.00652056989516479 ; 6 : 0.163579551221406 ; 7 : 0.0434395776822699 ;</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-25.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the left tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-26.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the right tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-27.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-28.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-29.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-30.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-31.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-32.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Example 5 : Number of Discoveries: 201 ; pihat0 = 0.3864133 
Log-likelihood with N(0, 2): -17602.8 
Log-likelihood with Gaussian Derivatives: -17607.36 
Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: 4.565327 
Normalized weights:
1 : -0.0149505230188178 ; 2 : 0.681006373173563 ; 3 : -0.029408092099831 ; 4 : 0.526597120212115 ; 5 : -0.0649823448928799 ; 6 : 0.248323484516014 ; 7 : -0.077154633635199 ;</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-33.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the left tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-34.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Zoom in to the right tail:</code></pre>
<p><img src="figure/alternative.rmd/unnamed-chunk-4-35.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-36.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-37.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-38.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-39.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative.rmd/unnamed-chunk-4-40.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.3.3 (2017-03-06)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS Sierra 10.12.5

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] cvxr_0.0.0.9009 EQL_1.0-0       ttutils_1.0-1   ashr_2.1-13    

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.10      knitr_1.16        magrittr_1.5     
 [4] REBayes_0.85      MASS_7.3-47       doParallel_1.0.10
 [7] pscl_1.4.9        SQUAREM_2016.10-1 lattice_0.20-35  
[10] R6_2.2.1          foreach_1.4.3     stringr_1.2.0    
[13] tools_3.3.3       parallel_3.3.3    grid_3.3.3       
[16] git2r_0.18.0      htmltools_0.3.6   iterators_1.0.8  
[19] assertthat_0.2.0  yaml_2.1.14       rprojroot_1.2    
[22] digest_0.6.12     Matrix_1.2-10     codetools_0.2-15 
[25] evaluate_0.10     rmarkdown_1.5     stringi_1.1.5    
[28] Rmosek_7.1.2      backports_1.0.5   truncnorm_1.0-7  </code></pre>
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