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<h1 class="title toc-ignore">Comparison with <code>Knockoff</code>: <span class="math inline">\(\Sigma_X\)</span> is Toeplitz</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-01-18</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-05-15</p>
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<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Applying CASH to linear regression variable selection, compared with other popular methods including BH, Knockoff.</p>
</div>
<div id="simulation-setting" class="section level2">
<h2>Simulation Setting</h2>
<p>The simulation setting is very similar to what’s used in Knockoff’s <a href="https://cran.r-project.org/web/packages/knockoff/vignettes/fixed.html">vignette</a>. The only notable difference is that the non-zero signals are normally distributed centered at zero, rather than constant.</p>
<p>Data are simulated by <span class="math display">\[
y_n = X_{n \times p}\beta_p + e_n
\]</span> where <span class="math display">\[
\begin{array}{c}
n = 2000 \\
p = 1000 \\
e_n \sim N(0, 1) \\
\beta \sim \eta\delta_0 + (1 - \eta)N(0, \sigma /\sqrt{n})
\end{array}
\]</span> Each row of <span class="math inline">\(X\)</span> is generated independently from a <span class="math inline">\(N(0, \Sigma_\rho)\)</span> distribution, where <span class="math inline">\(\left(\Sigma_\rho\right)_{j, k} = \rho^{|j - k|}\)</span>, a Toplitz matrix.</p>
<p>Every method selects the variables with respect to a nominal false discovery rate <span class="math inline">\(q = 0.1\)</span>.</p>
<p>In simulations, we are changing the values of the sparsity level <span class="math inline">\(\eta\)</span>, the signal strength <span class="math inline">\(\sigma\)</span>, the feature correlation <span class="math inline">\(\rho\)</span>.</p>
</div>
<div id="methods" class="section level2">
<h2>Methods</h2>
<ul>
<li><p>BH: First run multiple linear regression, then apply BH to obtained <span class="math inline">\(p\)</span>-values.</p></li>
<li><p>qvalue: First run multiple linear regression, then apply <code>qvalue::qvalue</code> to obtained <span class="math inline">\(p\)</span>-values.</p></li>
<li><p>Knockoff: Directly apply <code>knockoff::knockoff</code> on <span class="math inline">\(X\)</span>, <span class="math inline">\(y\)</span>.</p></li>
<li><p>ASH: First run multiple linear regression, then apply <code>ashr::ash</code> on obtained <span class="math inline">\(\hat\beta\)</span> and <span class="math inline">\(\hat{\text{se}}\left(\hat \beta\right)\)</span>, using normal mixture and normal likelihood.</p></li>
<li><p>CASH: First run multiple linear regression, then apply <code>cash</code> on obtained <span class="math inline">\(\hat\beta\)</span> and <span class="math inline">\(\hat{\text{se}}\left(\hat \beta\right)\)</span>, using normal mixture and normal likelihood, with default penalty on Gaussian derivative coefficients.</p></li>
<li><p>CASH+: CASH with perfect knowledge, using real noise level <span class="math inline">\(\text{se}\left(\hat{\beta}\right)\)</span>.</p></li>
</ul>
</div>
<div id="observations" class="section level2">
<h2>Observations</h2>
<ul>
<li><p>BH is very robust, very fast.</p></li>
<li><p>Knockoff is way too slow and way too conservative with signals being unimodal at zero. Perhaps it needs strong signals distinctly different from the “bulk.” Unimodal setting is really adversary to this method.</p></li>
<li><p><code>CASH</code> works fine, but not better than the basic <code>ASH</code>.</p></li>
</ul>
</div>
<div id="eta-0.9-sigma-5-rho-0.5" class="section level2">
<h2><span class="math inline">\(\eta = 0.9\)</span>, <span class="math inline">\(\sigma = 5\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<pre><code>No id variables; using all as measure variables</code></pre>
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</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-2-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-2-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-2-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>Warning in bplt(at[i], wid = width[i], stats = z$stats[, i], out = z$out[z
$group == : Outlier (-Inf) in boxplot 3 is not drawn</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-2-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-2-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-2-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-0.8-sigma-4-rho-0.5" class="section level2">
<h2><span class="math inline">\(\eta = 0.8\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-4-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-4-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>Warning in bplt(at[i], wid = width[i], stats = z$stats[, i], out = z$out[z
$group == : Outlier (-Inf) in boxplot 2 is not drawn</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-4-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-0.5-sigma-4-rho-0.5" class="section level2">
<h2><span class="math inline">\(\eta = 0.5\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-6-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-6-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/unnamed-chunk-6-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/unnamed-chunk-6-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-in-left0.5-0.6-0.7-0.8-0.9right-sigma-4-rho-0.5" class="section level2">
<h2><span class="math inline">\(\eta \in \left\{0.5, 0.6, 0.7, 0.8, 0.9\right\}\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<div id="overall-across-all-sparsity" class="section level3">
<h3>Overall across all sparsity</h3>
<p><img src="figure/knockoff.rmd/avg-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of avg-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/avg-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/avg-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of avg-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/avg-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-0.5-sigma-4-rho-0.5-1" class="section level3">
<h3><span class="math inline">\(\eta = 0.5\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 500-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/500-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 500-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/500-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/500-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 500-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/500-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-0.9-sigma-4-rho-0.5" class="section level3">
<h3><span class="math inline">\(\eta = 0.9\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 100-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/100-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 100-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/100-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/100-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 100-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/100-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="eta-in-left0.5-0.6-0.7-0.8-0.9right-sigma-3-rho-0.7" class="section level2">
<h2><span class="math inline">\(\eta \in \left\{0.5, 0.6, 0.7, 0.8, 0.9\right\}\)</span>, <span class="math inline">\(\sigma = 3\)</span>, <span class="math inline">\(\rho = 0.7\)</span></h2>
<div id="overall-across-all-sparsity-1" class="section level3">
<h3>Overall across all sparsity</h3>
<p><img src="figure/knockoff.rmd/avg%20weak-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of avg weak-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/avg weak-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/avg%20weak-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of avg weak-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/avg weak-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-0.5-sigma-3-rho-0.7" class="section level3">
<h3><span class="math inline">\(\eta = 0.5\)</span>, <span class="math inline">\(\sigma = 3\)</span>, <span class="math inline">\(\rho = 0.7\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500%20weak-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 500 weak-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/500 weak-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500%20weak-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 500 weak-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/500 weak-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/500%20weak-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 500 weak-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/500 weak-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="eta-0.9-sigma-3-rho-0.7" class="section level3">
<h3><span class="math inline">\(\eta = 0.9\)</span>, <span class="math inline">\(\sigma = 3\)</span>, <span class="math inline">\(\rho = 0.7\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100%20weak-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 100 weak-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/100 weak-1.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100%20weak-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 100 weak-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/100 weak-2.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff.rmd/100%20weak-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of 100 weak-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1c3ccea8098787267eec0fa7585a6ae6e16487e2/docs/figure/knockoff.rmd/100 weak-3.png" target="_blank">1c3ccea</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-01-25
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.4
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ggplot2_2.2.1 reshape2_1.4.3
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 knitr_1.20 whisker_0.3-2
[4] magrittr_1.5 workflowr_1.0.1 munsell_0.4.3
[7] colorspace_1.3-2 rlang_0.1.6 stringr_1.3.0
[10] plyr_1.8.4 tools_3.4.3 grid_3.4.3
[13] gtable_0.2.0 R.oo_1.21.0 git2r_0.21.0
[16] htmltools_0.3.6 lazyeval_0.2.1 yaml_2.1.18
[19] rprojroot_1.3-2 digest_0.6.15 tibble_1.4.1
[22] R.utils_2.6.0 evaluate_0.10.1 rmarkdown_1.9
[25] labeling_0.3 stringi_1.1.6 pillar_1.0.1
[28] compiler_3.4.3 scales_0.5.0 backports_1.1.2
[31] R.methodsS3_1.7.1</code></pre>
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