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As previously noted, that is an SNR = 0 case. Now we want to show how Gaussian derivatives work on cases with increasing SNRs.</p> <p>Here SNR in db is defined as <span class="math inline">\(10\log_{10}\left(\frac{\sigma_\beta^2}{\sigma_e^2}\right)\)</span>, where in our setting <span class="math inline">\(\sigma_e^2 = 1\)</span>, <span class="math inline">\(\sigma_z^2 = \sigma_\beta^2 + \sigma_e^2\)</span>.</p> </div> <div id="simulation-and-plots" class="section level2"> <h2>Simulation and plots</h2> <pre class="r"><code>library(ashr) source("../code/ecdfz.R")</code></pre> <pre><code>Warning: replacing previous import 'Matrix::crossprod' by 'gmp::crossprod' when loading 'cvxr'</code></pre> <pre><code>Warning: replacing previous import 'Matrix::tcrossprod' by 'gmp::tcrossprod' when loading 'cvxr'</code></pre> <pre class="r"><code>n = 1e4 m = 5 SNR.min = 1 SNR.max = 10 res = list() set.seed(777) for (SNR in SNR.min : SNR.max) { z.sd = sqrt(10^(SNR / 10) + 1) zmat = matrix(rnorm(n * m, 0, sd = z.sd), nrow = m, byrow = TRUE) res[[SNR]] = list() for (i in 1:m) { 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, firstk = TRUE) res[[SNR]][[i]] = list(z.sd = z.sd, z = z, p = p, bh.fd = bh.fd, pihat0.ash = pihat0.ash, ecdfz.fit = ecdfz.fit) } }</code></pre> <pre><code>## SNR = 1 ; sigma_z = 1.502972 ; True Distribution: N ( 0 , 2.258925 ) . Example 1 : SNR = 1 ; True Distribution: N ( 0 , 2.258925 ) ; Number of Discoveries: 428 ; pihat0 = 0.2709664 ; Log-likelihood by True Distribution N ( 0, 2.258925 ) : -18313.33 ; Log-likelihood by Gaussian Derivatives with K = 8 : -18318.05 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 8 : 4.720698 ; Normalized Weights of Gaussian Derivatives with K = 8 : 1 : -0.0138740386442539 ; 2 : 0.892281011245653 ; 3 : -0.028567009630833 ; 4 : 0.94253139322481 ; 5 : -0.0113795882900605 ; 6 : 0.809509538283857 ; 7 : 0.0229744434384983 ; 8 : 0.358299621647272 ; Log-likelihood by Gaussian Derivatives with K = 7 : -18348.81 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 7 : 35.47763 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0149703048359138 ; 2 : 0.866692877952966 ; 3 : -0.0301089228458965 ; 4 : 0.767046792878629 ; 5 : -0.0110444822556435 ; 6 : 0.380758391289584 ; 7 : 0.0254012390742264 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-6.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-7.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-8.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 1 ; True Distribution: N ( 0 , 2.258925 ) ; Number of Discoveries: 367 ; pihat0 = 0.2655976 ; Log-likelihood by True Distribution N ( 0, 2.258925 ) : -18229.62 ; Log-likelihood by Gaussian Derivatives with K = 9 : -18229.93 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 9 : 0.3054308 ; Normalized Weights of Gaussian Derivatives with K = 9 : 1 : 0.0109595744146466 ; 2 : 0.872494511525152 ; 3 : 0.0166435212643833 ; 4 : 0.834304921649212 ; 5 : -0.0172341620303621 ; 6 : 0.632087532212813 ; 7 : 0.00856994470962663 ; 8 : 0.246676236280596 ; 9 : 0.00960621718545015 ; Log-likelihood by Gaussian Derivatives with K = 8 : -18230.02 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 8 : 0.392468 ; Normalized Weights of Gaussian Derivatives with K = 8 : 1 : 0.0108635572606931 ; 2 : 0.872487253460917 ; 3 : 0.0151674590792571 ; 4 : 0.834395095058209 ; 5 : -0.0241729589279682 ; 6 : 0.632309611264995 ; 7 : -0.00494246114453377 ; 8 : 0.246809387644906 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-14.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-15.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-16.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 1 ; True Distribution: N ( 0 , 2.258925 ) ; Number of Discoveries: 414 ; pihat0 = 0.3111735 ; Log-likelihood by True Distribution N ( 0, 2.258925 ) : -18236.37 ; Log-likelihood by Gaussian Derivatives with K = 8 : -18234.66 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 8 : -1.715201 ; Normalized Weights of Gaussian Derivatives with K = 8 : 1 : -0.000572287615513397 ; 2 : 0.873484974327661 ; 3 : -0.0126862795911764 ; 4 : 0.891280588085513 ; 5 : 0.0448702002588981 ; 6 : 0.699292716557123 ; 7 : 0.0739176259383721 ; 8 : 0.269359515626305 ; Log-likelihood by Gaussian Derivatives with K = 7 : -18253.01 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 7 : 16.64117 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.00223200243361907 ; 2 : 0.853439452681122 ; 3 : -0.0206175539095362 ; 4 : 0.754777072315358 ; 5 : 0.0300355184347576 ; 6 : 0.37248254074971 ; 7 : 0.0629162247280905 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-18.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-21.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-22.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-23.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-24.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 1 ; True Distribution: N ( 0 , 2.258925 ) ; Number of Discoveries: 279 ; pihat0 = 0.2883564 ; Log-likelihood by True Distribution N ( 0, 2.258925 ) : -18180.99 ; Log-likelihood by Gaussian Derivatives with K = 7 : -18211.86 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 7 : 30.87149 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.00253181066300101 ; 2 : 0.835740744054015 ; 3 : -0.00828147519737005 ; 4 : 0.658405272063464 ; 5 : -0.0056181233262792 ; 6 : 0.279018029700389 ; 7 : 0.031205122505456 ; Log-likelihood by Gaussian Derivatives with K = 6 : -18217.69 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 6 : 36.69734 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.00342033433603602 ; 2 : 0.835461789922238 ; 3 : -0.0179069743937475 ; 4 : 0.658486690747026 ; 5 : -0.0365069966123492 ; 6 : 0.279269504553461 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-25.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-26.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-27.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-28.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-29.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-30.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-31.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-32.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 1 ; True Distribution: N ( 0 , 2.258925 ) ; Number of Discoveries: 408 ; pihat0 = 0.3458893 ; Log-likelihood by True Distribution N ( 0, 2.258925 ) : -18211.51 ; Log-likelihood by Gaussian Derivatives with K = 7 : -18217.85 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 7 : 6.346104 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0147557356837701 ; 2 : 0.847148023105142 ; 3 : -0.0300837913631588 ; 4 : 0.754691670792804 ; 5 : -0.0659841393438177 ; 6 : 0.381126272820956 ; 7 : -0.0839138576127453 ; Log-likelihood by Gaussian Derivatives with K = 6 : -18235.4 ; Log-likelihood Ratio between True Distribution N ( 0, 2.258925 ) and Fitted Gaussian Derivatives with K = 6 : 23.89227 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.010980181886604 ; 2 : 0.844912871994996 ; 3 : 0.00150880666789093 ; 4 : 0.75052104096923 ; 5 : 0.0233674006241364 ; 6 : 0.377820750277234 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-33.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-34.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-35.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-36.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-37.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-38.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-39.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-40.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 2 ; sigma_z = 1.60776 ; True Distribution: N ( 0 , 2.584893 ) . Example 1 : SNR = 2 ; True Distribution: N ( 0 , 2.584893 ) ; Number of Discoveries: 628 ; pihat0 = 0.2014281 ; Log-likelihood by True Distribution N ( 0, 2.584893 ) : -18883.1 ; Log-likelihood by Gaussian Derivatives with K = 8 : -18848.21 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 8 : -34.88926 ; Normalized Weights of Gaussian Derivatives with K = 8 : 1 : 0.0136147919601149 ; 2 : 1.08274316136917 ; 3 : 0.00337518683255761 ; 4 : 1.23442521358308 ; 5 : -0.0312064980495082 ; 6 : 1.12436528722487 ; 7 : -0.00506440306881227 ; 8 : 0.563409031383621 ; Log-likelihood by Gaussian Derivatives with K = 7 : -18958.36 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 7 : 75.26143 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : 0.0119440041009122 ; 2 : 1.03844441321769 ; 3 : 0.000649045654352769 ; 4 : 0.937019524556236 ; 5 : -0.0326509618902091 ; 6 : 0.419393818733931 ; 7 : -0.00443309470441701 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-41.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-42.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-43.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-44.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-45.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-46.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-47.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-48.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 2 ; True Distribution: N ( 0 , 2.584893 ) ; Number of Discoveries: 700 ; pihat0 = 0.2714964 ; Log-likelihood by True Distribution N ( 0, 2.584893 ) : -18940.46 ; Log-likelihood by Gaussian Derivatives with K = 7 : -19007.95 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 7 : 67.48755 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : 0.0164681394830972 ; 2 : 1.05636276082517 ; 3 : 0.0393323297755097 ; 4 : 1.05810403869031 ; 5 : 0.038214655867539 ; 6 : 0.528683713570606 ; 7 : -0.00290938755991068 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19007.97 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 6 : 67.50821 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.016590918892684 ; 2 : 1.05638106370673 ; 3 : 0.0404683327631348 ; 4 : 1.05816543499172 ; 5 : 0.04141481562775 ; 6 : 0.528741840837135 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-49.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-50.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-51.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-52.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-53.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-54.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-55.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-56.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 2 ; True Distribution: N ( 0 , 2.584893 ) ; Number of Discoveries: 727 ; pihat0 = 0.3134389 ; Log-likelihood by True Distribution N ( 0, 2.584893 ) : -18986.24 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19025.86 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 6 : 39.6199 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.034411297708113 ; 2 : 1.0798301969158 ; 3 : 0.0387605138305727 ; 4 : 1.09993225357172 ; 5 : 0.0189586458609482 ; 6 : 0.549984526433354 ; Log-likelihood by Gaussian Derivatives with K = 5 : -19213.48 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 5 : 227.2411 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0305257775976203 ; 2 : 0.961133806275765 ; 3 : 0.0266825322421371 ; 4 : 0.611958147781872 ; 5 : 0.0048237348092036 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-57.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-58.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-59.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-60.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-61.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-62.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-63.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-64.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 2 ; True Distribution: N ( 0 , 2.584893 ) ; Number of Discoveries: 754 ; pihat0 = 0.2578183 ; Log-likelihood by True Distribution N ( 0, 2.584893 ) : -18986.44 ; Log-likelihood by Gaussian Derivatives with K = 7 : -19023.87 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 7 : 37.42848 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0120449529453214 ; 2 : 1.08764707514979 ; 3 : -0.0272343286937293 ; 4 : 1.07640403751387 ; 5 : -0.0277378881145104 ; 6 : 0.530056762448694 ; 7 : 0.00998347331685409 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19024.22 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 6 : 37.77873 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.012497468175113 ; 2 : 1.08756111671169 ; 3 : -0.0313106614773657 ; 4 : 1.07622274974305 ; 5 : -0.0389318140481279 ; 6 : 0.529903478383342 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-65.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-66.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-67.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-68.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-69.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-70.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-71.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-72.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 2 ; True Distribution: N ( 0 , 2.584893 ) ; Number of Discoveries: 684 ; pihat0 = 0.2336033 ; Log-likelihood by True Distribution N ( 0, 2.584893 ) : -18910.99 ; Log-likelihood by Gaussian Derivatives with K = 7 : -18946.39 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 7 : 35.40402 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0128409389534885 ; 2 : 1.05565637812346 ; 3 : -0.0698680521778894 ; 4 : 1.00753208077044 ; 5 : -0.0136057563515818 ; 6 : 0.530753658180936 ; 7 : 0.0276611410399095 ; Log-likelihood by Gaussian Derivatives with K = 6 : -18956.21 ; Log-likelihood Ratio between True Distribution N ( 0, 2.584893 ) and Fitted Gaussian Derivatives with K = 6 : 45.22527 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0141250515327056 ; 2 : 1.05415473078761 ; 3 : -0.0806873609624797 ; 4 : 1.00529296660644 ; 5 : -0.0440799147371776 ; 6 : 0.529284900907757 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-73.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-74.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-75.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-76.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-77.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-78.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-79.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-80.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 3 ; sigma_z = 1.730683 ; True Distribution: N ( 0 , 2.995262 ) . Example 1 : SNR = 3 ; True Distribution: N ( 0 , 2.995262 ) ; Number of Discoveries: 1137 ; pihat0 = 0.2002261 ; Log-likelihood by True Distribution N ( 0, 2.995262 ) : -19749.88 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19874.22 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 6 : 124.3455 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0288992302921886 ; 2 : 1.30366812740725 ; 3 : 0.0628160596849958 ; 4 : 1.41969523193942 ; 5 : 0.0612663914524723 ; 6 : 0.736859567348341 ; Log-likelihood by Gaussian Derivatives with K = 5 : -20225.09 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 5 : 475.2066 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0238158009072628 ; 2 : 1.14465614448965 ; 3 : 0.0384889963108962 ; 4 : 0.757296526468208 ; 5 : 0.035015557729069 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-81.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-82.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-83.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-84.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-85.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-86.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-87.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-88.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 3 ; True Distribution: N ( 0 , 2.995262 ) ; Number of Discoveries: 1053 ; pihat0 = 0.236766 ; Log-likelihood by True Distribution N ( 0, 2.995262 ) : -19681.48 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19815.82 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 6 : 134.3427 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00318211711794664 ; 2 : 1.30247550307223 ; 3 : -0.0172188218881736 ; 4 : 1.42702661114986 ; 5 : -0.0363262272592029 ; 6 : 0.717683859109124 ; Log-likelihood by Gaussian Derivatives with K = 5 : -20085.13 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 5 : 403.6479 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0104814304575098 ; 2 : 1.13675011442731 ; 3 : 0.00188184844839905 ; 4 : 0.756404381575781 ; 5 : -0.0199154232069253 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-89.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-90.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-91.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-92.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-93.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-94.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-95.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-96.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 3 ; True Distribution: N ( 0 , 2.995262 ) ; Number of Discoveries: 1125 ; pihat0 = 0.2183353 ; Log-likelihood by True Distribution N ( 0, 2.995262 ) : -19799.13 ; Log-likelihood by Gaussian Derivatives with K = 7 : -19974.25 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 7 : 175.1193 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0369699199255488 ; 2 : 1.31265025712611 ; 3 : -0.107576867905454 ; 4 : 1.45773811185881 ; 5 : -0.121632069227899 ; 6 : 0.783548181004518 ; 7 : -0.0322240334306016 ; Log-likelihood by Gaussian Derivatives with K = 6 : -20032.16 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 6 : 233.0241 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0349918322709436 ; 2 : 1.30244269821314 ; 3 : -0.0928096088731791 ; 4 : 1.43703104429066 ; 5 : -0.0836064902501311 ; 6 : 0.768336876325314 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-97.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-98.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-99.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-100.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-101.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-102.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-103.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-104.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 3 ; True Distribution: N ( 0 , 2.995262 ) ; Number of Discoveries: 1015 ; pihat0 = 0.2005916 ; Log-likelihood by True Distribution N ( 0, 2.995262 ) : -19563.37 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19717.58 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 6 : 154.205 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0142014676711954 ; 2 : 1.23085121924937 ; 3 : 0.0193924773128272 ; 4 : 1.33650382371204 ; 5 : 0.00773080730718023 ; 6 : 0.714669370482247 ; Log-likelihood by Gaussian Derivatives with K = 5 : -20026.82 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 5 : 463.4474 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0138072992790593 ; 2 : 1.08230318475708 ; 3 : 0.0112953947087835 ; 4 : 0.70028252055515 ; 5 : -0.00401828618141428 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-105.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-106.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-107.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-108.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-109.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-110.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-111.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-112.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 3 ; True Distribution: N ( 0 , 2.995262 ) ; Number of Discoveries: 1116 ; pihat0 = 0.2059135 ; Log-likelihood by True Distribution N ( 0, 2.995262 ) : -19694.25 ; Log-likelihood by Gaussian Derivatives with K = 6 : -19875.35 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 6 : 181.0977 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0140242957481269 ; 2 : 1.27383551876828 ; 3 : -0.0464574751003468 ; 4 : 1.39777208974153 ; 5 : -0.0310039225932993 ; 6 : 0.75869513057903 ; Log-likelihood by Gaussian Derivatives with K = 5 : -20195.4 ; Log-likelihood Ratio between True Distribution N ( 0, 2.995262 ) and Fitted Gaussian Derivatives with K = 5 : 501.1484 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0132809857324255 ; 2 : 1.11756833715367 ; 3 : -0.0383825968226916 ; 4 : 0.718184548621535 ; 5 : -0.0246121563432862 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-113.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-114.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-115.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-116.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-117.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-118.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-119.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-120.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 4 ; sigma_z = 1.874003 ; True Distribution: N ( 0 , 3.511886 ) . Example 1 : SNR = 4 ; True Distribution: N ( 0 , 3.511886 ) ; Number of Discoveries: 1612 ; pihat0 = 0.2349198 ; Log-likelihood by True Distribution N ( 0, 3.511886 ) : -20505.62 ; Log-likelihood by Gaussian Derivatives with K = 7 : -20838.3 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 7 : 332.6796 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : 0.00263960278498244 ; 2 : 1.53936288484481 ; 3 : -0.000253103340590059 ; 4 : 1.90098025064903 ; 5 : 0.0266329163091332 ; 6 : 1.02003551948117 ; 7 : 0.024303389682097 ; Log-likelihood by Gaussian Derivatives with K = 6 : -20900.37 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 6 : 394.7513 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00155462847886588 ; 2 : 1.53004347413158 ; 3 : -0.0109051295788474 ; 4 : 1.8821764856015 ; 5 : -0.00189158567619316 ; 6 : 1.00635770850968 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-121.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-122.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-123.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-124.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-125.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-126.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-127.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-128.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 4 ; True Distribution: N ( 0 , 3.511886 ) ; Number of Discoveries: 1579 ; pihat0 = 0.1702099 ; Log-likelihood by True Distribution N ( 0, 3.511886 ) : -20538.23 ; Log-likelihood by Gaussian Derivatives with K = 6 : -20874.76 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 6 : 336.5326 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00714737291666616 ; 2 : 1.54777588924115 ; 3 : 0.0907515403200786 ; 4 : 1.86599937944303 ; 5 : 0.0724552423429854 ; 6 : 1.01923589785132 ; Log-likelihood by Gaussian Derivatives with K = 5 : -21468.18 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 5 : 929.9497 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00233712727950881 ; 2 : 1.30898534778199 ; 3 : 0.0614434451312987 ; 4 : 0.910461862499954 ; 5 : 0.05447641272924 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-129.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-130.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-131.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-132.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-133.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-134.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-135.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-136.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 4 ; True Distribution: N ( 0 , 3.511886 ) ; Number of Discoveries: 1537 ; pihat0 = 0.207689 ; Log-likelihood by True Distribution N ( 0, 3.511886 ) : -20375.11 ; Log-likelihood by Gaussian Derivatives with K = 6 : -20696.27 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 6 : 321.1552 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0129177627131297 ; 2 : 1.51027853149931 ; 3 : -0.0621604270315439 ; 4 : 1.7839515165522 ; 5 : -0.0463150827204791 ; 6 : 0.950489396221004 ; Log-likelihood by Gaussian Derivatives with K = 5 : -21160.44 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 5 : 785.3305 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0167322527670701 ; 2 : 1.28474315587364 ; 3 : -0.0507776149038532 ; 4 : 0.886307772175458 ; 5 : -0.0475668505725079 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-137.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-138.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-139.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-140.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-141.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-142.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-143.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-144.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 4 ; True Distribution: N ( 0 , 3.511886 ) ; Number of Discoveries: 1661 ; pihat0 = 0.2081909 ; Log-likelihood by True Distribution N ( 0, 3.511886 ) : -20619.85 ; Log-likelihood by Gaussian Derivatives with K = 6 : -21024.68 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 6 : 404.824 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.015109039889223 ; 2 : 1.58437283625333 ; 3 : 0.0261382021780695 ; 4 : 1.91545171554349 ; 5 : -0.00140914849247964 ; 6 : 1.00554828868042 ; Log-likelihood by Gaussian Derivatives with K = 5 : -21549.2 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 5 : 929.3471 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0216915779717695 ; 2 : 1.3390792645327 ; 3 : 0.0283549196023475 ; 4 : 0.961246767734933 ; 5 : -0.00869438265477574 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-145.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-146.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-147.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-148.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-149.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-150.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-151.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-152.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 4 ; True Distribution: N ( 0 , 3.511886 ) ; Number of Discoveries: 1539 ; pihat0 = 0.1875351 ; Log-likelihood by True Distribution N ( 0, 3.511886 ) : -20479.14 ; Log-likelihood by Gaussian Derivatives with K = 6 : -20822.78 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 6 : 343.6364 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0353091829171418 ; 2 : 1.5488252807078 ; 3 : 0.0815738367447848 ; 4 : 1.84626451120107 ; 5 : 0.0650913631373509 ; 6 : 0.984109496544227 ; Log-likelihood by Gaussian Derivatives with K = 5 : -21314.74 ; Log-likelihood Ratio between True Distribution N ( 0, 3.511886 ) and Fitted Gaussian Derivatives with K = 5 : 835.5978 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0275284735740842 ; 2 : 1.31280586018846 ; 3 : 0.0434390282435354 ; 4 : 0.909048058129576 ; 5 : 0.0225535285052623 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-153.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-154.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-155.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-156.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-157.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-158.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-159.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-160.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 5 ; sigma_z = 2.040166 ; True Distribution: N ( 0 , 4.162278 ) . Example 1 : SNR = 5 ; True Distribution: N ( 0 , 4.162278 ) ; Number of Discoveries: 2181 ; pihat0 = 0.1471061 ; Log-likelihood by True Distribution N ( 0, 4.162278 ) : -21456.54 ; Log-likelihood by Gaussian Derivatives with K = 7 : -22273.92 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 7 : 817.3892 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0277287247172747 ; 2 : 1.83729841608932 ; 3 : -0.0194061221798388 ; 4 : 2.38721144713735 ; 5 : 0.0669893025894355 ; 6 : 1.33833255575326 ; 7 : 0.0601856133000998 ; Log-likelihood by Gaussian Derivatives with K = 6 : -22316.92 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 6 : 860.386 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.031071765701026 ; 2 : 1.83200238307822 ; 3 : -0.0482607640760006 ; 4 : 2.3761386840482 ; 5 : -0.00660665954240478 ; 6 : 1.33000254948579 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-161.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-162.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-163.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-164.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-165.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-166.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-167.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-168.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 5 ; True Distribution: N ( 0 , 4.162278 ) ; Number of Discoveries: 2132 ; pihat0 = 0.1710206 ; Log-likelihood by True Distribution N ( 0, 4.162278 ) : -21360.19 ; Log-likelihood by Gaussian Derivatives with K = 7 : -22056.24 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 7 : 696.0522 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : -0.0302692305952895 ; 2 : 1.81095755971405 ; 3 : -0.123945606236119 ; 4 : 2.3625934945064 ; 5 : -0.16061012528612 ; 6 : 1.33954600169702 ; 7 : -0.0943016327906944 ; Log-likelihood by Gaussian Derivatives with K = 6 : -22150.52 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 6 : 790.3308 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0248164580551722 ; 2 : 1.79997755220376 ; 3 : -0.0786044913573935 ; 4 : 2.33958958474172 ; 5 : -0.0453793295236274 ; 6 : 1.32245550790188 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-169.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-170.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-171.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-172.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-173.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-174.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-175.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-176.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 5 ; True Distribution: N ( 0 , 4.162278 ) ; Number of Discoveries: 2022 ; pihat0 = 0.1179668 ; Log-likelihood by True Distribution N ( 0, 4.162278 ) : -21300.12 ; Log-likelihood by Gaussian Derivatives with K = 6 : -22024.14 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 6 : 724.023 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.000731727484443342 ; 2 : 1.77628066195772 ; 3 : -0.0183224310340538 ; 4 : 2.2544384353018 ; 5 : -0.0421725370426179 ; 6 : 1.25470467115244 ; Log-likelihood by Gaussian Derivatives with K = 5 : -22930.59 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 5 : 1630.466 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0109224327550832 ; 2 : 1.47160826685493 ; 3 : 0.0151877858144352 ; 4 : 1.05250752843448 ; 5 : -0.0104184675847151 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-177.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-178.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-179.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-180.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-181.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-182.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-183.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-184.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 5 ; True Distribution: N ( 0 , 4.162278 ) ; Number of Discoveries: 1955 ; pihat0 = 0.1390593 ; Log-likelihood by True Distribution N ( 0, 4.162278 ) : -21222.54 ; Log-likelihood by Gaussian Derivatives with K = 6 : -21916.99 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 6 : 694.4488 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.01082394354273 ; 2 : 1.76860322355297 ; 3 : -0.0212147026883038 ; 4 : 2.21317753779356 ; 5 : -0.0257310173508894 ; 6 : 1.20281481409197 ; Log-likelihood by Gaussian Derivatives with K = 5 : -22712.84 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 5 : 1490.302 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00519465956622556 ; 2 : 1.4699555974489 ; 3 : 0.000543992287838269 ; 4 : 1.04946806984471 ; 5 : -0.00321855532758913 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-185.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-186.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-187.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-188.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-189.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-190.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-191.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-192.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 5 ; True Distribution: N ( 0 , 4.162278 ) ; Number of Discoveries: 2084 ; pihat0 = 0.1694246 ; Log-likelihood by True Distribution N ( 0, 4.162278 ) : -21326.66 ; Log-likelihood by Gaussian Derivatives with K = 7 : -22026.65 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 7 : 699.9825 ; Normalized Weights of Gaussian Derivatives with K = 7 : 1 : 0.034611676260936 ; 2 : 1.8019192495016 ; 3 : 0.0702645299158879 ; 4 : 2.33324813754215 ; 5 : 0.0632684084249024 ; 6 : 1.30020235325766 ; 7 : 0.014351484665824 ; Log-likelihood by Gaussian Derivatives with K = 6 : -22108.35 ; Log-likelihood Ratio between True Distribution N ( 0, 4.162278 ) and Fitted Gaussian Derivatives with K = 6 : 781.6903 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0335621187013486 ; 2 : 1.79225356399466 ; 3 : 0.0627592982733057 ; 4 : 2.3134090885158 ; 5 : 0.0451433691894302 ; 6 : 1.28552689070843 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-193.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-194.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-195.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-196.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-197.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-198.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-199.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-200.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 6 ; sigma_z = 2.231831 ; True Distribution: N ( 0 , 4.981072 ) . Example 1 : SNR = 6 ; True Distribution: N ( 0 , 4.981072 ) ; Number of Discoveries: 2683 ; pihat0 = 0.1414556 ; Log-likelihood by True Distribution N ( 0, 4.981072 ) : -22239.22 ; Log-likelihood by Gaussian Derivatives with K = 6 : -23730.07 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 6 : 1490.851 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0296803542499545 ; 2 : 2.07614733637123 ; 3 : 0.0703932598614685 ; 4 : 2.85025980724701 ; 5 : 0.0596934102982105 ; 6 : 1.60294149075837 ; Log-likelihood by Gaussian Derivatives with K = 5 : -25033.34 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 5 : 2794.116 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.00787281872235832 ; 2 : 1.6598745095546 ; 3 : 0.0327468776020723 ; 4 : 1.26207912428607 ; 5 : 0.0409404655346192 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-201.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-202.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-203.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-204.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-205.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-206.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-207.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-208.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 6 ; True Distribution: N ( 0 , 4.981072 ) ; Number of Discoveries: 2782 ; pihat0 = 0.14379 ; Log-likelihood by True Distribution N ( 0, 4.981072 ) : -22307.73 ; Log-likelihood by Gaussian Derivatives with K = 6 : -23759.97 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 6 : 1452.239 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00968653765934509 ; 2 : 2.11764080114054 ; 3 : 0.00614427739560328 ; 4 : 2.91545496587689 ; 5 : -0.0290001943355776 ; 6 : 1.65319096959502 ; Log-likelihood by Gaussian Derivatives with K = 5 : -25154.82 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 5 : 2847.087 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0128279354380785 ; 2 : 1.68364560918076 ; 3 : 0.0160969375686653 ; 4 : 1.2704432431804 ; 5 : -0.0126453926481678 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-209.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-210.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-211.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-212.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-213.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-214.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-215.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-216.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 6 ; True Distribution: N ( 0 , 4.981072 ) ; Number of Discoveries: 2649 ; pihat0 = 0.1090815 ; Log-likelihood by True Distribution N ( 0, 4.981072 ) : -22258.28 ; Log-likelihood by Gaussian Derivatives with K = 6 : -23529.81 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 6 : 1271.533 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0011428411494417 ; 2 : 1.95662124625593 ; 3 : -0.0224171808790388 ; 4 : 2.49991702360509 ; 5 : -0.00366220135857955 ; 6 : 1.36757833770512 ; Log-likelihood by Gaussian Derivatives with K = 5 : -25155.11 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 5 : 2896.828 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00385085258044298 ; 2 : 1.67314628611184 ; 3 : -0.0169085908314758 ; 4 : 1.24156513591502 ; 5 : 0.00800909685413182 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-217.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-218.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-219.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-220.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-221.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-222.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-223.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-224.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 6 ; True Distribution: N ( 0 , 4.981072 ) ; Number of Discoveries: 2611 ; pihat0 = 0.1362864 ; Log-likelihood by True Distribution N ( 0, 4.981072 ) : -22094.39 ; Log-likelihood by Gaussian Derivatives with K = 5 : -24651.41 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 5 : 2557.016 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0207456250918557 ; 2 : 1.64776347409511 ; 3 : 0.0408372151048951 ; 4 : 1.23516153632877 ; 5 : -0.00209962997563955 ; Log-likelihood by Gaussian Derivatives with K = 4 : -24654.08 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 4 : 2559.692 ; Normalized Weights of Gaussian Derivatives with K = 4 : 1 : 0.0211027886566392 ; 2 : 1.64814189178002 ; 3 : 0.0425671978787183 ; 4 : 1.2356285923639 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-225.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-226.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-227.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-228.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-229.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-230.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-231.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-232.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 6 ; True Distribution: N ( 0 , 4.981072 ) ; Number of Discoveries: 2649 ; pihat0 = 0.1092557 ; Log-likelihood by True Distribution N ( 0, 4.981072 ) : -22215.9 ; Log-likelihood by Gaussian Derivatives with K = 6 : -23754.87 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 6 : 1538.97 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0116366532166927 ; 2 : 2.06025482639985 ; 3 : -0.011516485335873 ; 4 : 2.79783722718127 ; 5 : -0.0163523967952556 ; 6 : 1.58041329223154 ; Log-likelihood by Gaussian Derivatives with K = 5 : -25071.74 ; Log-likelihood Ratio between True Distribution N ( 0, 4.981072 ) and Fitted Gaussian Derivatives with K = 5 : 2855.842 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0167193129430813 ; 2 : 1.65088940488345 ; 3 : 0.0112739770964507 ; 4 : 1.23275554575975 ; 5 : 0.000220131488046469 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-233.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-234.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-235.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-236.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-237.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-238.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-239.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-240.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 7 ; sigma_z = 2.451912 ; True Distribution: N ( 0 , 6.011872 ) . Example 1 : SNR = 7 ; True Distribution: N ( 0 , 6.011872 ) ; Number of Discoveries: 3288 ; pihat0 = 0.08928676 ; Log-likelihood by True Distribution N ( 0, 6.011872 ) : -23260.7 ; Log-likelihood by Gaussian Derivatives with K = 6 : -26298.95 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 6 : 3038.251 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0183005878150952 ; 2 : 2.34484205485819 ; 3 : -0.0298238324987321 ; 4 : 3.36367097754695 ; 5 : -0.0263635043049669 ; 6 : 1.95573111007512 ; Log-likelihood by Gaussian Derivatives with K = 5 : -28345.43 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 5 : 5084.722 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0207339186941601 ; 2 : 1.82651582430218 ; 3 : -0.0233716676525867 ; 4 : 1.40075833244072 ; 5 : -0.016455641490395 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-241.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-242.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-243.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-244.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-245.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-246.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-247.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-248.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 7 ; True Distribution: N ( 0 , 6.011872 ) ; Number of Discoveries: 3333 ; pihat0 = 0.09588348 ; Log-likelihood by True Distribution N ( 0, 6.011872 ) : -23213.25 ; Log-likelihood by Gaussian Derivatives with K = 6 : -25965.71 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 6 : 2752.465 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0157497749468068 ; 2 : 2.29522111448563 ; 3 : -0.0504594888083231 ; 4 : 3.2716206983689 ; 5 : -0.0434371740666126 ; 6 : 1.88971138582365 ; Log-likelihood by Gaussian Derivatives with K = 5 : -28063.64 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 5 : 4850.391 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00125856771424838 ; 2 : 1.82848294685367 ; 3 : -0.00862514412274113 ; 4 : 1.41531531673069 ; 5 : -0.0113003931850965 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-249.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-250.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-251.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-252.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-253.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-254.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-255.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-256.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 7 ; True Distribution: N ( 0 , 6.011872 ) ; Number of Discoveries: 3176 ; pihat0 = 0.1297791 ; Log-likelihood by True Distribution N ( 0, 6.011872 ) : -23073.36 ; Log-likelihood by Gaussian Derivatives with K = 6 : -25730.54 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 6 : 2657.185 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00547516103293227 ; 2 : 2.24562303322161 ; 3 : 0.0493165965305574 ; 4 : 3.16335068855266 ; 5 : 0.0400002165361984 ; 6 : 1.7738638592201 ; Log-likelihood by Gaussian Derivatives with K = 5 : -27667.42 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 5 : 4594.065 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00781400536548428 ; 2 : 1.78824512444825 ; 3 : 0.0305422935251163 ; 4 : 1.41139215177903 ; 5 : 0.0299925365790375 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-257.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-258.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-259.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-260.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-261.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-262.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-263.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-264.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 7 ; True Distribution: N ( 0 , 6.011872 ) ; Number of Discoveries: 3194 ; pihat0 = 0.1000719 ; Log-likelihood by True Distribution N ( 0, 6.011872 ) : -23103.79 ; Log-likelihood by Gaussian Derivatives with K = 6 : -25884.9 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 6 : 2781.118 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.019831930425411 ; 2 : 2.28723915997433 ; 3 : 0.0334406444701634 ; 4 : 3.25360120421447 ; 5 : 0.0113572300675825 ; 6 : 1.87558158578094 ; Log-likelihood by Gaussian Derivatives with K = 5 : -27793.23 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 5 : 4689.439 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.021214778576848 ; 2 : 1.80481775316164 ; 3 : 0.0330643574613194 ; 4 : 1.38231477760038 ; 5 : 0.00396533588188546 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-265.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-266.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-267.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-268.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-269.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-270.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-271.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-272.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 7 ; True Distribution: N ( 0 , 6.011872 ) ; Number of Discoveries: 3235 ; pihat0 = 0.1134862 ; Log-likelihood by True Distribution N ( 0, 6.011872 ) : -23044.41 ; Log-likelihood by Gaussian Derivatives with K = 6 : -25484.16 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 6 : 2439.756 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00715356253823022 ; 2 : 2.27436645331824 ; 3 : 0.00105952221137104 ; 4 : 3.20013131956896 ; 5 : 0.00118057804423982 ; 6 : 1.78681955956309 ; Log-likelihood by Gaussian Derivatives with K = 5 : -27372.31 ; Log-likelihood Ratio between True Distribution N ( 0, 6.011872 ) and Fitted Gaussian Derivatives with K = 5 : 4327.9 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0105083874094317 ; 2 : 1.8119786043687 ; 3 : 0.0202691225905926 ; 4 : 1.41690219054158 ; 5 : 0.0194692182232266 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-273.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-274.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-275.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-276.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-277.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-278.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-279.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-280.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 8 ; sigma_z = 2.703622 ; True Distribution: N ( 0 , 7.309573 ) . Example 1 : SNR = 8 ; True Distribution: N ( 0 , 7.309573 ) ; Number of Discoveries: 3768 ; pihat0 = 0.07033265 ; Log-likelihood by True Distribution N ( 0, 7.309573 ) : -24083.99 ; Log-likelihood by Gaussian Derivatives with K = 6 : -28842.7 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 6 : 4758.717 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.000677264290150885 ; 2 : 2.53874231778277 ; 3 : -0.00561937268793838 ; 4 : 3.72454384154009 ; 5 : 0.0210042933210722 ; 6 : 2.16093057958714 ; Log-likelihood by Gaussian Derivatives with K = 5 : -31542.48 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 5 : 7458.492 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00719984954595982 ; 2 : 1.97534018546734 ; 3 : -0.0159195281654859 ; 4 : 1.54212762351448 ; 5 : 0.0134058274996992 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-281.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-282.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-283.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-284.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-285.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-286.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-287.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-288.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 8 ; True Distribution: N ( 0 , 7.309573 ) ; Number of Discoveries: 3950 ; pihat0 = 0.08759456 ; Log-likelihood by True Distribution N ( 0, 7.309573 ) : -24216.05 ; Log-likelihood by Gaussian Derivatives with K = 6 : -29100.75 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 6 : 4884.702 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00471661063579975 ; 2 : 2.52271547230107 ; 3 : -0.0328067689142773 ; 4 : 3.70553494414676 ; 5 : -0.0498341710748214 ; 6 : 2.15158211636193 ; Log-likelihood by Gaussian Derivatives with K = 5 : -32137.13 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 5 : 7921.075 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.022164827818991 ; 2 : 1.97427468128317 ; 3 : 0.0092283343891227 ; 4 : 1.56649329906334 ; 5 : -0.0252294737613793 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-289.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-290.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-291.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-292.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-293.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-294.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-295.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-296.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 8 ; True Distribution: N ( 0 , 7.309573 ) ; Number of Discoveries: 3869 ; pihat0 = 0.07653203 ; Log-likelihood by True Distribution N ( 0, 7.309573 ) : -24190.71 ; Log-likelihood by Gaussian Derivatives with K = 6 : -29333.05 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 6 : 5142.335 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.00362625059133025 ; 2 : 2.59051580056953 ; 3 : -0.0110301347352026 ; 4 : 3.85770452188801 ; 5 : -0.0325210274551719 ; 6 : 2.26162663690691 ; Log-likelihood by Gaussian Derivatives with K = 5 : -32114.77 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 5 : 7924.057 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.00823123809677517 ; 2 : 1.97684485306676 ; 3 : 0.0108132563951897 ; 4 : 1.54891889152428 ; 5 : -0.0162193203234067 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-297.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-298.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-299.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-300.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-301.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-302.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-303.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-304.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 8 ; True Distribution: N ( 0 , 7.309573 ) ; Number of Discoveries: 3952 ; pihat0 = 0.08656298 ; Log-likelihood by True Distribution N ( 0, 7.309573 ) : -24187 ; Log-likelihood by Gaussian Derivatives with K = 6 : -29045.38 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 6 : 4858.372 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0206034448835079 ; 2 : 2.56735543697531 ; 3 : -0.0195051816112572 ; 4 : 3.80522687316745 ; 5 : -0.00201564696868051 ; 6 : 2.22210637473464 ; Log-likelihood by Gaussian Derivatives with K = 5 : -31944.19 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 5 : 7757.19 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.027016205328516 ; 2 : 1.98489748843074 ; 3 : -0.0270749603534199 ; 4 : 1.56565594649678 ; 5 : -0.00611079194533588 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-305.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-306.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-307.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-308.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-309.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-310.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-311.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-312.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 8 ; True Distribution: N ( 0 , 7.309573 ) ; Number of Discoveries: 3899 ; pihat0 = 0.1077967 ; Log-likelihood by True Distribution N ( 0, 7.309573 ) : -24166.42 ; Log-likelihood by Gaussian Derivatives with K = 6 : -29164.99 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 6 : 4998.566 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.0427398762094977 ; 2 : 2.61806859203705 ; 3 : 0.0777731701832885 ; 4 : 3.91059345316949 ; 5 : 0.0448564315430896 ; 6 : 2.26738457298974 ; Log-likelihood by Gaussian Derivatives with K = 5 : -31851.62 ; Log-likelihood Ratio between True Distribution N ( 0, 7.309573 ) and Fitted Gaussian Derivatives with K = 5 : 7685.195 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.045548659776884 ; 2 : 1.98322811808131 ; 3 : 0.0659398154220372 ; 4 : 1.58559580543908 ; 5 : 0.0182038967082177 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-313.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-314.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-315.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-316.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-317.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-318.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-319.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-320.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 9 ; sigma_z = 2.990532 ; True Distribution: N ( 0 , 8.943282 ) . Example 1 : SNR = 9 ; True Distribution: N ( 0 , 8.943282 ) ; Number of Discoveries: 4482 ; pihat0 = 0.07214675 ; Log-likelihood by True Distribution N ( 0, 8.943282 ) : -25260.2 ; Log-likelihood by Gaussian Derivatives with K = 6 : -33591.66 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 6 : 8331.452 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0200952702053063 ; 2 : 2.70577564712737 ; 3 : 0.0234755479272126 ; 4 : 4.00771362970997 ; 5 : 0.0207216012591999 ; 6 : 2.29917777268476 ; Log-likelihood by Gaussian Derivatives with K = 5 : -37680.12 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 5 : 12419.92 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0355092626574237 ; 2 : 2.12868510144083 ; 3 : -0.0328340929155301 ; 4 : 1.71968825181984 ; 5 : -0.0155997016077485 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-321.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-322.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-323.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-324.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-325.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-326.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-327.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-328.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 9 ; True Distribution: N ( 0 , 8.943282 ) ; Number of Discoveries: 4331 ; pihat0 = 0.06865425 ; Log-likelihood by True Distribution N ( 0, 8.943282 ) : -25049.88 ; Log-likelihood by Gaussian Derivatives with K = 6 : -32879.85 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 6 : 7829.973 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0118481448044448 ; 2 : 2.7796986362372 ; 3 : 0.0147953186401661 ; 4 : 4.24114709990809 ; 5 : 0.0157825845679687 ; 6 : 2.50268448455396 ; Log-likelihood by Gaussian Derivatives with K = 5 : -36530.56 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 5 : 11480.68 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0396370307801417 ; 2 : 2.09691444712674 ; 3 : -0.0244298920072881 ; 4 : 1.66843078627915 ; 5 : 0.0167443134840248 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-329.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-330.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-331.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-332.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-333.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-334.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-335.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-336.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 9 ; True Distribution: N ( 0 , 8.943282 ) ; Number of Discoveries: 4546 ; pihat0 = 0.08109916 ; Log-likelihood by True Distribution N ( 0, 8.943282 ) : -25164.49 ; Log-likelihood by Gaussian Derivatives with K = 6 : -32855.24 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 6 : 7690.759 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0370511982981982 ; 2 : 2.72329041198641 ; 3 : -0.127103598366047 ; 4 : 4.0766935735135 ; 5 : -0.0976827927260967 ; 6 : 2.35066412909989 ; Log-likelihood by Gaussian Derivatives with K = 5 : -36872.49 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 5 : 11708.01 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0140648455733178 ; 2 : 2.1339367625933 ; 3 : -0.0494389492890455 ; 4 : 1.72841919139732 ; 5 : -0.0335277614382204 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-337.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-338.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-339.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-340.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-341.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-342.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-343.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-344.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 9 ; True Distribution: N ( 0 , 8.943282 ) ; Number of Discoveries: 4492 ; pihat0 = 0.08146913 ; Log-likelihood by True Distribution N ( 0, 8.943282 ) : -25203.14 ; Log-likelihood by Gaussian Derivatives with K = 6 : -33373.18 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 6 : 8170.031 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00501567898026982 ; 2 : 2.87465931120391 ; 3 : 0.00290829147950967 ; 4 : 4.44704795480167 ; 5 : 0.0132047958198434 ; 6 : 2.6190712213443 ; Log-likelihood by Gaussian Derivatives with K = 5 : -37233.33 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 5 : 12030.19 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.00413829110193593 ; 2 : 2.11361129838446 ; 3 : -0.0107119022734637 ; 4 : 1.71057016325765 ; 5 : -0.00346874900298212 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-345.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-346.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-347.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-348.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-349.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-350.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-351.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-352.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 9 ; True Distribution: N ( 0 , 8.943282 ) ; Number of Discoveries: 4466 ; pihat0 = 0.08603856 ; Log-likelihood by True Distribution N ( 0, 8.943282 ) : -25165.91 ; Log-likelihood by Gaussian Derivatives with K = 6 : -33165.64 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 6 : 7999.728 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00596118834903648 ; 2 : 2.76422782520799 ; 3 : -0.013950817489483 ; 4 : 4.20491448828645 ; 5 : -0.0290508365341806 ; 6 : 2.4568157323229 ; Log-likelihood by Gaussian Derivatives with K = 5 : -37088.68 ; Log-likelihood Ratio between True Distribution N ( 0, 8.943282 ) and Fitted Gaussian Derivatives with K = 5 : 11922.77 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : 0.0132826541910612 ; 2 : 2.10603816814406 ; 3 : -0.000943359578635805 ; 4 : 1.70381670500689 ; 5 : -0.0249672044643002 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-353.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-354.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-355.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-356.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-357.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-358.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-359.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-360.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>## SNR = 10 ; sigma_z = 3.316625 ; True Distribution: N ( 0 , 11 ) . Example 1 : SNR = 10 ; True Distribution: N ( 0 , 11 ) ; Number of Discoveries: 5011 ; pihat0 = 0.07150713 ; Log-likelihood by True Distribution N ( 0, 11 ) : -26181.19 ; Log-likelihood by Gaussian Derivatives with K = 6 : -38850.85 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 6 : 12669.66 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.0274236929352572 ; 2 : 2.89711168724025 ; 3 : -0.0223603095778489 ; 4 : 4.45623541443822 ; 5 : 0.00101775045562966 ; 6 : 2.62862143174931 ; Log-likelihood by Gaussian Derivatives with K = 5 : -44143.81 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 5 : 17962.61 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0385032350077447 ; 2 : 2.2210137346095 ; 3 : -0.0447036615903929 ; 4 : 1.81344530778662 ; 5 : -0.00202440280451175 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-361.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-362.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-363.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-364.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-365.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-366.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-367.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-368.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 2 : SNR = 10 ; True Distribution: N ( 0 , 11 ) ; Number of Discoveries: 4990 ; pihat0 = 0.07062845 ; Log-likelihood by True Distribution N ( 0, 11 ) : -26216.58 ; Log-likelihood by Gaussian Derivatives with K = 6 : -39023.06 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 6 : 12806.48 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : 0.00188608241211791 ; 2 : 2.80497119845542 ; 3 : 0.012664085793072 ; 4 : 4.19662730932605 ; 5 : 0.00702606774451777 ; 6 : 2.39432486209987 ; Log-likelihood by Gaussian Derivatives with K = 5 : -44506.92 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 5 : 18290.34 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0139504523290885 ; 2 : 2.22195423539989 ; 3 : -0.0107302268163738 ; 4 : 1.82807934181925 ; 5 : 0.00109459089469162 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-369.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-370.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-371.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-372.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-373.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-374.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-375.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-376.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 3 : SNR = 10 ; True Distribution: N ( 0 , 11 ) ; Number of Discoveries: 5017 ; pihat0 = 0.07467248 ; Log-likelihood by True Distribution N ( 0, 11 ) : -26207.72 ; Log-likelihood by Gaussian Derivatives with K = 5 : -44357.02 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 5 : 18149.3 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.000959029668663561 ; 2 : 2.22010699044865 ; 3 : -0.0135059729501484 ; 4 : 1.82792253239188 ; 5 : -0.0192050739456098 ; Log-likelihood by Gaussian Derivatives with K = 4 : -44357.86 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 4 : 18150.14 ; Normalized Weights of Gaussian Derivatives with K = 4 : 1 : 0.00215492288428279 ; 2 : 2.22008711070275 ; 3 : 0.00282896641163462 ; 4 : 1.82788088385223 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-377.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-378.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-379.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-380.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-381.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-382.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-383.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-384.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 4 : SNR = 10 ; True Distribution: N ( 0 , 11 ) ; Number of Discoveries: 5066 ; pihat0 = 0.0705596 ; Log-likelihood by True Distribution N ( 0, 11 ) : -26197.01 ; Log-likelihood by Gaussian Derivatives with K = 6 : -39090.32 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 6 : 12893.31 ; Normalized Weights of Gaussian Derivatives with K = 6 : 1 : -0.00369120704032752 ; 2 : 3.12154933079295 ; 3 : -0.00365745072147679 ; 4 : 4.9296943766698 ; 5 : 0.00379751707834492 ; 6 : 2.90730030596267 ; Log-likelihood by Gaussian Derivatives with K = 5 : -43985.54 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 5 : 17788.53 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.00978192326482823 ; 2 : 2.25189648665813 ; 3 : -0.00479443179567208 ; 4 : 1.84445913631995 ; 5 : 0.0167719092761371 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-385.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-386.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-387.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-388.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-389.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-390.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-391.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-392.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Example 5 : SNR = 10 ; True Distribution: N ( 0 , 11 ) ; Number of Discoveries: 4924 ; pihat0 = 0.08210577 ; Log-likelihood by True Distribution N ( 0, 11 ) : -26103.44 ; Log-likelihood by Gaussian Derivatives with K = 5 : -43511.18 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 5 : 17407.74 ; Normalized Weights of Gaussian Derivatives with K = 5 : 1 : -0.0169044216000957 ; 2 : 2.21803447163974 ; 3 : -0.0170247648783184 ; 4 : 1.82679005604263 ; 5 : -0.000477147476056258 ; Log-likelihood by Gaussian Derivatives with K = 4 : -43511.18 ; Log-likelihood Ratio between True Distribution N ( 0, 11 ) and Fitted Gaussian Derivatives with K = 4 : 17407.74 ; Normalized Weights of Gaussian Derivatives with K = 4 : 1 : -0.0168231555661593 ; 2 : 2.21803546066004 ; 3 : -0.01661630451235 ; 4 : 1.82679077771681 ; </code></pre> <p><img src="figure/alternative2.rmd/plot-393.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-394.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-395.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-396.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-397.png" width="672" style="display: block; margin: auto;" /><img src="figure/alternative2.rmd/plot-398.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the left tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-399.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Zoom in to the right tail:</code></pre> <p><img src="figure/alternative2.rmd/plot-400.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="conclusion" class="section level2"> <h2>Conclusion</h2> <p>As soon as SNR is increasing, the Gaussian derivatives stop fitting well. The weights don’t <a href="gaussian_derivatives_5.html#weight_constraints">look reasonable</a>, and the fitting in tails is especially bad.</p> <p>It indicates that this method can identify deviation of the empirical distribution from the standard normal caused by correlated null from that caused by true effects when the effects are large.</p> <p>By the way, BH procedure seems very sensible in this task. It gives a lot of discoveries even when SNR = 1 that are too many to be credibly seen as from correlated null. Even SNR = 0 case, <a href="alternative.html">BH arguably gives more than expected discoveries</a>, which indicates it’s more likely to be true effects rather than correlation.</p> <p><strong>So at this moment, we should concentrate on larger effects, whereas keep the SNR = 1 case, as well as BH’s “surprisingly good performance” in mind.</strong></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 3.4.2 (2017-09-28) Platform: x86_64-apple-darwin15.6.0 (64-bit) Running under: macOS Sierra 10.12.6 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] cvxr_0.0.0.9400 EQL_1.0-0 ttutils_1.0-1 ashr_2.1-27 loaded via a namespace (and not attached): [1] Rcpp_0.12.13 knitr_1.17 magrittr_1.5 [4] MASS_7.3-47 doParallel_1.0.11 pscl_1.5.2 [7] SQUAREM_2017.10-1 lattice_0.20-35 foreach_1.4.3 [10] stringr_1.2.0 tools_3.4.2 parallel_3.4.2 [13] grid_3.4.2 git2r_0.19.0 htmltools_0.3.6 [16] iterators_1.0.8 yaml_2.1.14 rprojroot_1.2 [19] digest_0.6.12 gmp_0.5-13.1 Matrix_1.2-11 [22] codetools_0.2-15 evaluate_0.10.1 rmarkdown_1.6 [25] stringi_1.1.5 compiler_3.4.2 backports_1.1.1 [28] 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'; // 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://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"; document.getElementsByTagName("head")[0].appendChild(script); })(); </script> </body> </html>