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Can we fit real signals with Gaussian derivatives with reasonable weights? Let’s start with the most basic 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> </div> <div id="illustration" class="section level2"> <h2>Illustration</h2> <p>Let normalized weights <span class="math inline">\(W_k^s = W_k\sqrt{k!}\)</span> with variance <span class="math inline">\(\text{var}(W_k^s) = \alpha_k = \bar{\rho_{ij}^k}\)</span>.</p> <pre class="r"><code>n = 1e4 m = 5 set.seed(777) zmat = matrix(rnorm(n * m, 0, sd = sqrt(2)), nrow = m)</code></pre> <pre class="r"><code>library(ashr) source("../code/ecdfz.R") res = list() for (i in 1:3) { z = zmat[i, ] p = (1 - pnorm(abs(z))) * 2 bh.fd = sum(p.adjust(p, method = "BH") <= 0.05) pihat0.ash = get_pi0(ash(z, 1, method = "fdr")) 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>Number of Discoveries: 207 ; pihat0 = 0.3309512 Log-likelihood with N(0, 2): -17629.27 Log-likelihood with Gaussian Derivatives: -17634.06 Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: 4.790331 Normalized weights: 1 : -0.0177035771860527 ; 2 : 0.689887136368602 ; 3 : -0.0509478089605657 ; 4 : 0.474815572091947 ; 5 : -0.0255292356243276 ; 6 : 0.197260355735445 ; 7 : 0.0209642526780779 ;</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 right tails:</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 left tails:</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>Number of Discoveries: 210 ; pihat0 = 0.3484933 Log-likelihood with N(0, 2): -17695.54 Log-likelihood with Gaussian Derivatives: -17691.84 Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: -3.698983 Normalized weights: 1 : -0.00662167797406594 ; 2 : 0.716277504382886 ; 3 : -0.0310338589748257 ; 4 : 0.594319021051848 ; 5 : -0.00405607696274195 ; 6 : 0.364148970093227 ; 7 : 0.0347791611900783 ; 8 : 0.0862488827136366 ;</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 right tails:</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 left tails:</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>Number of Discoveries: 190 ; pihat0 = 0.3438938 Log-likelihood with N(0, 2): -17620.41 Log-likelihood with Gaussian Derivatives: -17620.47 Log-likelihood ratio between true N(0, 2) and fitted Gaussian derivatives: 0.05853415 Normalized weights: 1 : 0.00412988146644346 ; 2 : 0.691757356275142 ; 3 : 0.0624186506260473 ; 4 : 0.566017599686127 ; 5 : 0.0720236790272216 ; 6 : 0.359005476183793 ; 7 : 0.0491535934289797 ; 8 : 0.11765212650001 ;</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 right tails:</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 left tails:</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> </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 3.3.2 (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 other attached packages: [1] cvxr_0.0.0.9009 EQL_1.0-0 ttutils_1.0-1 ashr_2.1.5 loaded via a namespace (and not attached): [1] Rcpp_0.12.10 knitr_1.15.1 magrittr_1.5 [4] MASS_7.3-45 doParallel_1.0.10 pscl_1.4.9 [7] SQUAREM_2016.10-1 lattice_0.20-34 foreach_1.4.3 [10] stringr_1.1.0 tools_3.3.2 parallel_3.3.2 [13] grid_3.3.2 htmltools_0.3.5 iterators_1.0.8 [16] yaml_2.1.14 rprojroot_1.2 digest_0.6.9 [19] Matrix_1.2-7.1 codetools_0.2-15 evaluate_0.10 [22] rmarkdown_1.3 stringi_1.1.2 backports_1.0.5 [25] truncnorm_1.0-7 </code></pre> </div> <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'; 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