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{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-27T07:45:08","documenter_version":"1.2.1"}}
{"documenter":{"julia_version":"1.10.0","generation_timestamp":"2024-01-27T08:03:04","documenter_version":"1.2.1"}}
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1.5</code></pre><p>Create a 1d array of alpha parameter penalties values that determine the penalties mix between <span>$L_1$</span> and <span>$L_2$</span> to fit the estimates according to the Elastic Net penalization method. In the case of Lasso regression (<span>$L_1$</span> regularization), alpha should be 1, and 0 for Ridge regression (<span>$L_2$</span> regularization). If the alphas are not in descending order, they will be automatically sorted by <code>mlmnet</code>.</p><pre><code class="language-julia hljs">alphas = reverse(collect(0:0.5:1));</code></pre><p>If <code>alphas</code> argument is omitted, a Lasso regression will be applied which is equivalent to <code>alphas = [1]</code>. </p><h3 id="Elastic-Net-penalization-algorithms-selection"><a class="docs-heading-anchor" href="#Elastic-Net-penalization-algorithms-selection">Elastic Net penalization algorithms selection</a><a id="Elastic-Net-penalization-algorithms-selection-1"></a><a class="docs-heading-anchor-permalink" href="#Elastic-Net-penalization-algorithms-selection" title="Permalink"></a></h3><p>The algorithms available for fitting Elastic Net penalized estimates in <code>mlmnet</code> function come with customizable keyword arguments for fine-tuning. The <code>method</code> keyword argument selects the function implementing the Elastic-net penalty estimation method. The default method is <strong>ista</strong>; alternative options include <strong>fista</strong>, <strong>fista_bt</strong>, <strong>admm</strong>, and <strong>cd</strong>. <em>Note: Any irrelevant arguments will simply be disregarded.</em></p><table><tr><th style="text-align: right">Algorithm</th><th style="text-align: right">Methods</th><th style="text-align: right">Parameter</th><th style="text-align: right">Default</th><th style="text-align: right">Description</th></tr><tr><td style="text-align: right"><strong>Coordinate Descent</strong></td><td style="text-align: right">&quot;cd&quot;</td><td style="text-align: right"><code>isRandom</code></td><td style="text-align: right">true</td><td style="text-align: right">Determines the use of either random or cyclic updates</td></tr><tr><td style="text-align: right"><strong>Active Coordinate Descent</strong></td><td style="text-align: right">&quot;cd_active&quot;</td><td style="text-align: right"><code>isRandom</code></td><td style="text-align: right">true</td><td style="text-align: right">Specifies the choice between random and cyclic updates</td></tr><tr><td style="text-align: right"><strong>ISTA</strong> with fixed step size</td><td style="text-align: right">&quot;ista&quot;</td><td style="text-align: right"><code>stepsize</code></td><td style="text-align: right">0.01</td><td style="text-align: right">Sets a fixed step size for updates</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>setStepsize</code></td><td style="text-align: right">true</td><td style="text-align: right">Decides if the fixed step size is to be computed, overriding <code>stepsize</code></td></tr><tr><td style="text-align: right"><strong>FISTA</strong> with fixed step size</td><td style="text-align: right">&quot;fista&quot;</td><td style="text-align: right"><code>stepsize</code></td><td style="text-align: right">0.01</td><td style="text-align: right">Establishes a fixed step size for updates</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>setStepsize</code></td><td style="text-align: right">true</td><td style="text-align: right">Determines if the fixed step size should be recalculated, overriding <code>stepsize</code></td></tr><tr><td style="text-align: right"><strong>FISTA</strong> with backtracking</td><td style="text-align: right">&quot;fista_bt&quot;</td><td style="text-align: right"><code>stepsize</code></td><td style="text-align: right">0.01</td><td style="text-align: right">Indicates the initial step size for updates</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>gamma</code></td><td style="text-align: right">0.5</td><td style="text-align: right">The multiplier for step size during backtracking/line search</td></tr><tr><td style="text-align: right"><strong>ADMM</strong></td><td style="text-align: right">&quot;admm&quot;</td><td style="text-align: right"><code>rho</code></td><td style="text-align: right">1.0</td><td style="text-align: right">The parameter influencing ADMM tuning</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>setRho</code></td><td style="text-align: right">true</td><td style="text-align: right">Decides whether the ADMM tuning parameter <code>rho</code> is to be auto-calculated</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>tau_incr</code></td><td style="text-align: right">2.0</td><td style="text-align: right">The factor for increasing the ADMM tuning parameter</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>tau_decr</code></td><td style="text-align: right">2.0</td><td style="text-align: right">The factor for decreasing the ADMM tuning parameter</td></tr><tr><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><code>mu</code></td><td style="text-align: right">10.0</td><td style="text-align: right">The parameter influencing the balance between primal and dual residuals</td></tr></table><h3 id="Estimation"><a class="docs-heading-anchor" href="#Estimation">Estimation</a><a id="Estimation-1"></a><a class="docs-heading-anchor-permalink" href="#Estimation" title="Permalink"></a></h3><p>In this example, we will use the &#39;fista&#39; method for our estimation process. Given that our design matrix already incorporates an intercept, we specify that there is no need to add an additional intercept to the design matrices <strong>X</strong> and <strong>Z</strong>.</p><pre><code class="language-julia hljs">est_fista = mlmnet(
1.5</code></pre><p>Create a 1d array of alpha parameter penalties values that determine the penalties mix between <span>$L_1$</span> and <span>$L_2$</span> to fit the estimates according to the Elastic Net penalization method. In the case of Lasso regression (<span>$L_1$</span> regularization), alpha should be 1, and 0 for Ridge regression (<span>$L_2$</span> regularization). If the alphas are not in descending order, they will be automatically sorted by <code>mlmnet</code>.</p><pre><code class="language-julia hljs">alphas = reverse(collect(0:0.5:1));</code></pre><p>If <code>alphas</code> argument is omitted, a Lasso regression will be applied which is equivalent to <code>alphas = [1]</code>. </p><h3 id="Elastic-Net-penalization-algorithms-selection"><a class="docs-heading-anchor" href="#Elastic-Net-penalization-algorithms-selection">Elastic Net penalization algorithms selection</a><a id="Elastic-Net-penalization-algorithms-selection-1"></a><a class="docs-heading-anchor-permalink" href="#Elastic-Net-penalization-algorithms-selection" title="Permalink"></a></h3><p>The algorithms available for fitting Elastic Net penalized estimates in <code>mlmnet</code> function come with customizable keyword arguments for fine-tuning. The <code>method</code> keyword argument selects the function implementing the Elastic-net penalty estimation method. The default method is <strong>ista</strong>; alternative options include <strong>fista</strong>, <strong>fista_bt</strong>, <strong>admm</strong>, and <strong>cd</strong>. <em>Note: Any irrelevant arguments will simply be disregarded.</em></p><table><tr><th style="text-align: left">Algorithm</th><th style="text-align: right">Methods</th><th style="text-align: right">Parameter</th><th style="text-align: right">Default</th><th style="text-align: left">Description</th></tr><tr><td style="text-align: left"><strong>Coordinate Descent</strong></td><td style="text-align: right">&quot;cd&quot;</td><td style="text-align: right"><code>isRandom</code></td><td style="text-align: right">true</td><td style="text-align: left">Determines the use of either random or cyclic updates</td></tr><tr><td style="text-align: left"><strong>Active Coordinate Descent</strong></td><td style="text-align: right">&quot;cd_active&quot;</td><td style="text-align: right"><code>isRandom</code></td><td style="text-align: right">true</td><td style="text-align: left">Specifies the choice between random and cyclic updates</td></tr><tr><td style="text-align: left"><strong>ISTA</strong> with fixed step size</td><td style="text-align: right">&quot;ista&quot;</td><td style="text-align: right"><code>stepsize</code></td><td style="text-align: right">0.01</td><td style="text-align: left">Sets a fixed step size for updates</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>setStepsize</code></td><td style="text-align: right">true</td><td style="text-align: left">Decides if the fixed step size is to be computed, overriding <code>stepsize</code></td></tr><tr><td style="text-align: left"><strong>FISTA</strong> with fixed step size</td><td style="text-align: right">&quot;fista&quot;</td><td style="text-align: right"><code>stepsize</code></td><td style="text-align: right">0.01</td><td style="text-align: left">Establishes a fixed step size for updates</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>setStepsize</code></td><td style="text-align: right">true</td><td style="text-align: left">Determines if the fixed step size should be recalculated, overriding <code>stepsize</code></td></tr><tr><td style="text-align: left"><strong>FISTA</strong> with backtracking</td><td style="text-align: right">&quot;fista_bt&quot;</td><td style="text-align: right"><code>stepsize</code></td><td style="text-align: right">0.01</td><td style="text-align: left">Indicates the initial step size for updates</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>gamma</code></td><td style="text-align: right">0.5</td><td style="text-align: left">The multiplier for step size during backtracking/line search</td></tr><tr><td style="text-align: left"><strong>ADMM</strong></td><td style="text-align: right">&quot;admm&quot;</td><td style="text-align: right"><code>rho</code></td><td style="text-align: right">1.0</td><td style="text-align: left">The parameter influencing ADMM tuning</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>setRho</code></td><td style="text-align: right">true</td><td style="text-align: left">Decides whether the ADMM tuning parameter <code>rho</code> is to be auto-calculated</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>tau_incr</code></td><td style="text-align: right">2.0</td><td style="text-align: left">The factor for increasing the ADMM tuning parameter</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>tau_decr</code></td><td style="text-align: right">2.0</td><td style="text-align: left">The factor for decreasing the ADMM tuning parameter</td></tr><tr><td style="text-align: left"></td><td style="text-align: right"></td><td style="text-align: right"><code>mu</code></td><td style="text-align: right">10.0</td><td style="text-align: left">The parameter influencing the balance between primal and dual residuals</td></tr></table><h3 id="Estimation"><a class="docs-heading-anchor" href="#Estimation">Estimation</a><a id="Estimation-1"></a><a class="docs-heading-anchor-permalink" href="#Estimation" title="Permalink"></a></h3><p>In this example, we will use the &#39;fista&#39; method for our estimation process. Given that our design matrix already incorporates an intercept, we specify that there is no need to add an additional intercept to the design matrices <strong>X</strong> and <strong>Z</strong>.</p><pre><code class="language-julia hljs">est_fista = mlmnet(
dat,
[lambdas[1]], [alphas[1]],
method = &quot;fista&quot;,
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size = (800, 300)
),
title = [&quot;Residuals&quot; &quot;Distribution of the residuals&quot;]
)</code></pre><p><img src="../images/output_resid_hist.svg" alt="svg"/></p><p>Additional details can be found in the documentation for specific functions.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../functions/">Types and Functions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Saturday 27 January 2024 07:45">Saturday 27 January 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
)</code></pre><p><img src="../images/output_resid_hist.svg" alt="svg"/></p><p>Additional details can be found in the documentation for specific functions.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../functions/">Types and Functions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Saturday 27 January 2024 08:03">Saturday 27 January 2024</span>. Using Julia version 1.10.0.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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