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<h1 class="title toc-ignore">A detailed study of Fixed-<span class="math inline">\(X\)</span> and Model-<span class="math inline">\(X\)</span> knockoffs</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-03-01</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-05-15</p>
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<p><summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/LSun/truncash/tree/388e65e06000e313c170a82f3ed57346f6024897" target="_blank">388e65e</a> </summary></p>
Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated. <br><br> Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use <code>wflow_publish</code> or <code>wflow_git_commit</code>). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
<pre><code>
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</code></pre>
Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
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Lei Sun
</td>
<td style="text-align:left;">
2018-03-01
</td>
<td style="text-align:left;">
correlations
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/9d66e7250774642b015ae61d08be22a353339d28/docs/knockoff_8.html" target="_blank">9d66e72</a>
</td>
<td style="text-align:left;">
Lei Sun
</td>
<td style="text-align:left;">
2018-03-01
</td>
<td style="text-align:left;">
correlations
</td>
</tr>
</tbody>
</table>
</ul>
</details>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>The idea in <code>Knockoff</code> is to generate knockoff variables <span class="math inline">\(\tilde X\)</span> which 1) keep the relationship of original variables <span class="math inline">\(X\)</span> 2) are unlike the original variables 3) are null.</p>
<div id="for-fixed-x-knockoff" class="section level3">
<h3>For Fixed-<span class="math inline">\(X\)</span> Knockoff</h3>
<p><span class="math display">\[
\begin{array}{c}
\tilde X_i^T\tilde X_i = X_i^TX_i \\
\tilde X_i^T\tilde X_j = X_i^T \tilde X_j = \tilde X_i^T X_j = X_i^TX_j \\
\left|\tilde X_i^T X_i\right|: \text{as small as possible}
\end{array}
\]</span></p>
<p>The first two constraints are to control the Type I error, whereas the third is to increase power. Furthermore, there are two methods to make <span class="math inline">\(\left|\tilde X_i^T X_i\right|\)</span> as small as possible.</p>
<ul>
<li>Equi: <span class="math inline">\(\tilde X_1^T X_1 = \tilde X_2^T X_2 = \cdots = \tilde X_p^T X_p \geq 0\)</span> as close to zero as possible.</li>
<li>SDP: <span class="math inline">\(\tilde X_1^T X_1 + \tilde X_2^T X_2 + \cdots + \tilde X_p^T X_p\)</span> as close to zero as possible with <span class="math inline">\(\tilde X_i^T X_i \geq 0\)</span>.</li>
</ul>
</div>
<div id="for-model-x-knockoff" class="section level3">
<h3>For Model-X Knockoff</h3>
<p><span class="math display">\[
\begin{array}{c}
Var(X_i) = Var(\tilde X_i) \\
cor(\tilde X_i, \tilde X_j) = cor(X_i, \tilde X_j)
= cor(\tilde X_i, X_j) = cor(X_i, X_j) \\
\left|cor(\tilde X_i, X_i)\right|: \text{as small as possible}
\end{array}
\]</span></p>
<p>Similarly, the first two constraints are to control the Type I error, whereas the third is to increase power. Furthermore, there are three methods to make <span class="math inline">\(\left|cor(\tilde X_i, X_i)\right|\)</span> as small as possible.</p>
<ul>
<li>Equi: <span class="math inline">\(cor(\tilde X_1, X_1) = cor(\tilde X_2, X_2) = \cdots = cor(\tilde X_p, X_p) \geq 0\)</span> as close to zero as possible.</li>
<li>SDP: <span class="math inline">\(\left|cor(\tilde X_1, X_1)\right| + \left|cor(\tilde X_2, X_2)\right| + \cdots + \left|cor(\tilde X_p, X_p)\right|\)</span> as small as possible.</li>
<li>aSDP: An approximation to SDP to speed it up and also retain the power to some extend.</li>
</ul>
<p>After centering <span class="math inline">\(X_j\)</span>’s, the constraints on the Fix-<span class="math inline">\(X\)</span> design can be seen as the constraints on the <strong>sample correlation</strong> structure, whereas those on the Model-<span class="math inline">\(X\)</span> design as on the <strong>population correlation</strong> structure. It’s no wonder the Model-<span class="math inline">\(X\)</span> design is believed to be more relaxed, and hence, more powerful.</p>
<pre class="r"><code>n <- 3000
p <- 1000
k <- 50
m <- 100
q <- 0.1</code></pre>
</div>
</div>
<div id="scenario-1-independent-gaussian-columns" class="section level2">
<h2>Scenario 1: Independent Gaussian Columns</h2>
<p><span class="math display">\[
X_{ij} \overset{iid}{\sim} N\left(0, \left(\frac{1}{\sqrt n}\right)^2\right)
\]</span></p>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
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Author
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Date
</th>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-3-1.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-3-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-3-2.png:</em></summary>
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</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/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-3-2.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-3-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-3-3.png:</em></summary>
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</th>
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-3-3.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-3-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-3-4.png:</em></summary>
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<th style="text-align:left;">
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</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/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-3-4.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-3-4.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-3-5.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-3-5.png:</em></summary>
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<th style="text-align:left;">
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</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/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-3-5.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<div id="fixed-design" class="section level3">
<h3>Fixed Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
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</th>
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-4-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-4-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-4-1.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
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</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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-4-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-4-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-4-2.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
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</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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-4-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-4-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-4-3.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-design" class="section level3">
<h3>Model-<span class="math inline">\(X\)</span> Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-1.png:</em></summary>
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<th style="text-align:left;">
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</th>
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-5-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-5-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-5-1.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-2.png:</em></summary>
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<th style="text-align:left;">
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</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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-5-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-5-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-5-2.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-5-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-3.png:</em></summary>
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<tr>
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</th>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-5-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-5-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-5-3.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="scenario-2-equal-correlation" class="section level2">
<h2>Scenario 2: Equal-correlation</h2>
<p><span class="math display">\[
\begin{array}{c}
X_{ij} \sim N\left(0, \left(\frac{1}{\sqrt n}\right)^2\right) \\
cor(X_{ij}, X_{ij'}) \equiv \rho
\end{array}
\]</span></p>
<div id="rho-0.5" class="section level3">
<h3><span class="math inline">\(\rho = 0.5\)</span></h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
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</thead>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-6-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-6-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-6-1.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-6-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-6-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-6-2.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.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;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-6-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-6-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-6-3.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-6-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-4.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-6-4.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-6-4.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-6-5.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-5.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-6-5.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<div id="fixed-design-1" class="section level4">
<h4>Fixed Design</h4>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-7-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-7-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-7-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-7-1.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-7-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-7-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-7-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-7-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-7-2.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-7-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-7-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-7-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-7-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/7b679107f47eefb511aa2fbab55570edcfecb99f/docs/figure/knockoff_8.rmd/unnamed-chunk-7-3.png" target="_blank">7b67910</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-02
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-design-1" class="section level4">
<h4>Model-<span class="math inline">\(X\)</span> Design</h4>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-8-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-8-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-8-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-8-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-8-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-8-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-8-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-8-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-8-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-8-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-8-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="rho-0.99" class="section level3">
<h3><span class="math inline">\(\rho = 0.99\)</span></h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-9-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-9-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-9-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-9-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-9-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-9-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-9-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-9-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-9-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-9-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-9-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-9-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-9-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-9-4.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-9-4.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-9-4.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-9-5.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-9-5.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-9-5.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-9-5.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<div id="fixed-design-2" class="section level4">
<h4>Fixed Design</h4>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-10-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-10-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-10-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-10-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-10-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-10-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-10-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-10-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-design-2" class="section level4">
<h4>Model-<span class="math inline">\(X\)</span> Design</h4>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-11-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-11-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-11-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-11-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-11-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-11-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-11-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-11-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-11-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-11-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-11-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-11-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
</div>
<div id="scenario-3-extreme-local-correlation" class="section level2">
<h2>Scenario 3: Extreme local correlation</h2>
<p><span class="math display">\[
\begin{array}{c}
X_{ij} \sim N\left(0, \left(\frac{1}{\sqrt n}\right)^2\right) \\
cor(X_{i1}, X_{i2}) = cor(X_{i3}, X_{i4}) = \cdots = cor(X_{i(p-1)}, X_{ip}) = \rho = 0.99
\end{array}
\]</span></p>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-12-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-12-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-12-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-12-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-12-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-12-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-12-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-12-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-12-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-12-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-12-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-12-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-12-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-12-4.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-12-4.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-12-5.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-12-5.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-12-5.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<div id="fixed-design-3" class="section level3">
<h3>Fixed Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-13-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-13-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-13-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-13-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-13-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-13-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-13-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-13-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-13-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-13-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-13-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-13-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-design-3" class="section level3">
<h3>Model-<span class="math inline">\(X\)</span> Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-14-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-14-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-14-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-14-1.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-14-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-14-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-14-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-14-2.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-14-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-14-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-14-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/480666b20f31cf8df366bbf2ee4613819fb2a503/docs/figure/knockoff_8.rmd/unnamed-chunk-14-3.png" target="_blank">480666b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="scenario-4-factor-model-for-x" class="section level2">
<h2>Scenario 4: Factor Model for <span class="math inline">\(X\)</span></h2>
<p><span class="math display">\[
\begin{array}{c}
\text{each row }X_{i}^T \sim N\left(0, \frac{1}n\Sigma_{p}\right) \\
\Sigma_p = \texttt{cov2cor}(B_{p \times d}B_{d\times p}^T + I)\\
B_{ij} \overset{iid}{\sim} N(0, 1)
\end{array}
\]</span></p>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-15-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-15-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-15-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-15-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-15-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-15-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-15-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-15-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-15-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-15-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-15-4.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-15-4.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-15-5.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-15-5.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-15-5.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<div id="fixed-design-4" class="section level3">
<h3>Fixed Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-16-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-16-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-16-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-16-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-16-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-16-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-16-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-16-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-16-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-design-4" class="section level3">
<h3>Model-<span class="math inline">\(X\)</span> Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-17-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-17-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-17-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-17-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-17-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-17-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-17-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-17-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-17-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="scenario-5-factor-model-for-hatbeta" class="section level2">
<h2>Scenario 5: Factor Model for <span class="math inline">\(\hat\beta\)</span></h2>
<p><span class="math display">\[
\begin{array}{c}
\text{each row }X_{i}^T \sim N\left(0, \frac{1}n\Sigma_{p}\right) \\
\Sigma_p = \texttt{cov2cor}(\left(B_{p \times d}B_{d\times p}^T + I\right)^{-1})\\
B_{ij} \overset{iid}{\sim} N(0, 1)
\end{array}
\]</span></p>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-18-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-18-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-18-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-18-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-18-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-18-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-18-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-18-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-18-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-18-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-18-4.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-18-4.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-18-5.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-18-5.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-18-5.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<div id="fixed-design-5" class="section level3">
<h3>Fixed Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-19-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-19-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-19-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-19-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-19-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-19-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-19-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-19-3.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-19-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-design-5" class="section level3">
<h3>Model-<span class="math inline">\(X\)</span> Design</h3>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-20-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-20-1.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-20-1.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-20-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-20-2.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/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-20-2.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_8.rmd/unnamed-chunk-20-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-20-3.png:</em></summary>
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<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/d3d259c1f0776da3eae685089542ecf58c0f22b6/docs/figure/knockoff_8.rmd/unnamed-chunk-20-3.png" target="_blank">d3d259c</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-10
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="observation" class="section level2">
<h2>Observation</h2>
<p>Model-<span class="math inline">\(X\)</span> knockoff indeed increases power substantially.</p>
<ul>
<li><p>Reason 1: it builds better knockoffs as they are less similar to originals, since Model-<span class="math inline">\(X\)</span> is about population correlation not sample correlation.</p></li>
<li><p>Reason 2: Model-<span class="math inline">\(X\)</span> unleashes a large swarm of complicated test statistics which are not applicable for Fixed-<span class="math inline">\(X\)</span>.</p></li>
</ul>
<p>However, for that particular case when we have <span class="math display">\[\Sigma_\hat\beta\]</span> as a factor model, Model-<span class="math inline">\(X\)</span> knockoffs are still not good.</p>
</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] parallel stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] lattice_0.20-35 doMC_1.3.5 iterators_1.0.9 foreach_1.4.4
[5] ggplot2_2.2.1 reshape2_1.4.3 Matrix_1.2-12 knockoff_0.3.0
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 compiler_3.4.3 pillar_1.0.1
[4] git2r_0.21.0 plyr_1.8.4 workflowr_1.0.1
[7] R.methodsS3_1.7.1 R.utils_2.6.0 tools_3.4.3
[10] digest_0.6.15 evaluate_0.10.1 tibble_1.4.1
[13] gtable_0.2.0 rlang_0.1.6 yaml_2.1.18
[16] stringr_1.3.0 knitr_1.20 rprojroot_1.3-2
[19] grid_3.4.3 rmarkdown_1.9 magrittr_1.5
[22] whisker_0.3-2 backports_1.1.2 scales_0.5.0
[25] codetools_0.2-15 htmltools_0.3.6 colorspace_1.3-2
[28] labeling_0.3 stringi_1.1.6 lazyeval_0.2.1
[31] munsell_0.4.3 R.oo_1.21.0 </code></pre>
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