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We are using <a href="gaussian_derivatives_2.html">an automated rule</a>, yet sometimes the objective (log-likelihood) could fail to pass the optimal criterion before the optimization becomes unstable.</p> <p>Recall that we make <a href="gaussian_derivatives.html">two key assumptions</a> to make the problem tractable. <em>In place of the original second constraint of non-negativity, we use <span class="math inline">\(n\)</span> observed <span class="math inline">\(z\)</span> scores instead of all <span class="math inline">\(x\in\mathbb{R}\)</span>.</em> Therefore, when <span class="math inline">\(K\)</span> gets larger, and the higher order Gaussian derivatives involved get more complicated, it’s possible that the optimal solution will satisfy the non-negativity constraint for all <span class="math inline">\(n\)</span> observed <span class="math inline">\(z\)</span> scores, <strong>but not the whole real line.</strong> <em>This issue also happens to <a href="gaussian_derivatives_2.html">some well-behaved examples</a> if looked closely.</em></p> <p>Meanwhile, sometimes an optimal <span class="math inline">\(\hat K\)</span> can be found according to <a href="gaussian_derivatives_2.html">the rule</a>, but it looks like overfitting. A <span class="math inline">\(K < \hat K\)</span> appears better.</p> <p>Here we have two examples.</p> <pre class="r"><code>source("../code/ecdfz.R")</code></pre> <pre class="r"><code>z = read.table("../output/z_null_liver_777.txt") p = read.table("../output/p_null_liver_777.txt")</code></pre> <pre class="r"><code>library(ashr) DataSet = c(522, 724) res_DataSet = list() for (i in 1:length(DataSet)) { zscore = as.numeric(z[DataSet[i], ]) fit.ecdfz = ecdfz.optimal(zscore) fit.ash = ash(zscore, 1, method = "fdr") fit.ash.pi0 = get_pi0(fit.ash) pvalue = as.numeric(p[DataSet[i], ]) fd.bh = sum(p.adjust(pvalue, method = "BH") <= 0.05) res_DataSet[[i]] = list(DataSet = DataSet[i], fit.ecdfz = fit.ecdfz, fit.ash = fit.ash, fit.ash.pi0 = fit.ash.pi0, fd.bh = fd.bh, zscore = zscore, pvalue = pvalue) }</code></pre> <pre class="r"><code>library(EQL) x.pt = seq(-5, 5, 0.01) H.pt = sapply(1:15, EQL::hermite, x = x.pt)</code></pre> </div> <div id="example-1-numerical-instability-when-k-is-too-large" class="section level2"> <h2>Example 1: Numerical instability when <span class="math inline">\(K\)</span> is too large</h2> <pre><code>Data Set 724 : Number of BH's False Discoveries: 79 ; ASH's pihat0 = 0.01606004</code></pre> <p><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /></p> <p>In this example <a href="gaussian_derivatives_2.html">the automated rule</a> fails to find an optimal <span class="math inline">\(\hat K\)</span>. Note that the fitted log-likelihood increased until seemingly reached a plateau, but didn’t quite make the cut. After that, as <span class="math inline">\(K\)</span> keeps getting larger, the optimization becomes unstable. The blue <span class="math inline">\(K = 14\)</span> line obviously breaks <a href="gaussian_derivatives.html">the non-negativity constraint</a> for <span class="math inline">\(x \neq z_i\)</span>, the <span class="math inline">\(n\)</span> observed <span class="math inline">\(z\)</span> scores.</p> </div> <div id="example-2-overfitting-when-k-is-larger-than-what-appears-necessary" class="section level2"> <h2>Example 2: Overfitting when <span class="math inline">\(K\)</span> is larger than what appears necessary</h2> <pre><code>Data Set 522 : Number of BH's False Discoveries: 4 ; ASH's pihat0 = 0.02083846</code></pre> <p><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gaussian_derivatives_3.rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /></p> <p>In this example <a href="gaussian_derivatives_2.html">the automated rule</a> is able to find an optimal <span class="math inline">\(\hat K = 8\)</span>. However, the green <span class="math inline">\(K = 6\)</span> lines seems better visually. Their difference in the fitted log-likelihood is very small, although larger than what the rule requires.</p> </div> <div id="conclusion" class="section level2"> <h2>Conclusion</h2> <p>Things can go very wrong when the number of fitted Gaussian derivatives <span class="math inline">\(K\)</span> is too large, and it implies that we cannot blindly fit ever growing <span class="math inline">\(K\)</span> and hope the fitted log-likelihood converges. On the other hand, the good news is oftentimes we can still reach a pattern of increasing log-likelihoods, which gives a reasonable <span class="math inline">\(K\)</span>, before the optimization becomes unstable, although it might be not the optimal <span class="math inline">\(K\)</span> we would find by the current log-likelihood ratio test motivated rule.</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 loaded via a namespace (and not attached): [1] Rcpp_0.12.10 lattice_0.20-34 digest_0.6.11 rprojroot_1.2 [5] MASS_7.3-45 grid_3.3.2 backports_1.0.5 magrittr_1.5 [9] evaluate_0.10 stringi_1.1.2 Matrix_1.2-7.1 rmarkdown_1.3 [13] tools_3.3.2 stringr_1.2.0 yaml_2.1.14 htmltools_0.3.5 [17] knitr_1.15.1 </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'; // 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> --> </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://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"; document.getElementsByTagName("head")[0].appendChild(script); })(); </script> </body> </html>