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id="sklearn.linear_model.lasso_path"> <span class="sig-prename descclassname"><span class="pre">sklearn.linear_model.</span></span><span class="sig-name descname"><span class="pre">lasso_path</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.001</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_alphas</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">100</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">alphas</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">precompute</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'auto'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Xy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">copy_X</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">coef_init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">return_n_iter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">positive</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">params</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/linear_model/_coordinate_descent.py#L184"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.linear_model.lasso_path" title="Link to this definition">#</a></dt> <dd><p>Compute Lasso path with coordinate descent.</p> <p>The Lasso optimization function varies for mono and multi-outputs.</p> <p>For mono-output tasks it is:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n_samples</span><span class="p">))</span> <span class="o">*</span> <span class="o">||</span><span class="n">y</span> <span class="o">-</span> <span class="n">Xw</span><span class="o">||^</span><span class="mi">2_2</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="o">||</span><span class="n">w</span><span class="o">||</span><span class="n">_1</span> </pre></div> </div> <p>For multi-output tasks it is:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n_samples</span><span class="p">))</span> <span class="o">*</span> <span class="o">||</span><span class="n">Y</span> <span class="o">-</span> <span class="n">XW</span><span class="o">||^</span><span class="mi">2</span><span class="n">_Fro</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">*</span> <span class="o">||</span><span class="n">W</span><span class="o">||</span><span class="n">_21</span> </pre></div> </div> <p>Where:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="o">||</span><span class="n">W</span><span class="o">||</span><span class="n">_21</span> <span class="o">=</span> \<span class="n">sum_i</span> \<span class="n">sqrt</span><span class="p">{</span>\<span class="n">sum_j</span> <span class="n">w_</span><span class="p">{</span><span class="n">ij</span><span class="p">}</span><span class="o">^</span><span class="mi">2</span><span class="p">}</span> </pre></div> </div> <p>i.e. the sum of norm of each row.</p> <p>Read more in the <a class="reference internal" href="../linear_model.html#lasso"><span class="std std-ref">User Guide</span></a>.</p> <dl class="field-list simple"> <dt class="field-odd">Parameters<span class="colon">:</span></dt> <dd class="field-odd"><dl class="simple"> <dt><strong>X</strong><span class="classifier">{array-like, sparse matrix} of shape (n_samples, n_features)</span></dt><dd><p>Training data. Pass directly as Fortran-contiguous data to avoid unnecessary memory duplication. If <code class="docutils literal notranslate"><span class="pre">y</span></code> is mono-output then <code class="docutils literal notranslate"><span class="pre">X</span></code> can be sparse.</p> </dd> <dt><strong>y</strong><span class="classifier">{array-like, sparse matrix} of shape (n_samples,) or (n_samples, n_targets)</span></dt><dd><p>Target values.</p> </dd> <dt><strong>eps</strong><span class="classifier">float, default=1e-3</span></dt><dd><p>Length of the path. <code class="docutils literal notranslate"><span class="pre">eps=1e-3</span></code> means that <code class="docutils literal notranslate"><span class="pre">alpha_min</span> <span class="pre">/</span> <span class="pre">alpha_max</span> <span class="pre">=</span> <span class="pre">1e-3</span></code>.</p> </dd> <dt><strong>n_alphas</strong><span class="classifier">int, default=100</span></dt><dd><p>Number of alphas along the regularization path.</p> </dd> <dt><strong>alphas</strong><span class="classifier">array-like, default=None</span></dt><dd><p>List of alphas where to compute the models. If <code class="docutils literal notranslate"><span class="pre">None</span></code> alphas are set automatically.</p> </dd> <dt><strong>precompute</strong><span class="classifier">‘auto’, bool or array-like of shape (n_features, n_features), default=’auto’</span></dt><dd><p>Whether to use a precomputed Gram matrix to speed up calculations. If set to <code class="docutils literal notranslate"><span class="pre">'auto'</span></code> let us decide. The Gram matrix can also be passed as argument.</p> </dd> <dt><strong>Xy</strong><span class="classifier">array-like of shape (n_features,) or (n_features, n_targets), default=None</span></dt><dd><p>Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed.</p> </dd> <dt><strong>copy_X</strong><span class="classifier">bool, default=True</span></dt><dd><p>If <code class="docutils literal notranslate"><span class="pre">True</span></code>, X will be copied; else, it may be overwritten.</p> </dd> <dt><strong>coef_init</strong><span class="classifier">array-like of shape (n_features, ), default=None</span></dt><dd><p>The initial values of the coefficients.</p> </dd> <dt><strong>verbose</strong><span class="classifier">bool or int, default=False</span></dt><dd><p>Amount of verbosity.</p> </dd> <dt><strong>return_n_iter</strong><span class="classifier">bool, default=False</span></dt><dd><p>Whether to return the number of iterations or not.</p> </dd> <dt><strong>positive</strong><span class="classifier">bool, default=False</span></dt><dd><p>If set to True, forces coefficients to be positive. (Only allowed when <code class="docutils literal notranslate"><span class="pre">y.ndim</span> <span class="pre">==</span> <span class="pre">1</span></code>).</p> </dd> <dt><strong>**params</strong><span class="classifier">kwargs</span></dt><dd><p>Keyword arguments passed to the coordinate descent solver.</p> </dd> </dl> </dd> <dt class="field-even">Returns<span class="colon">:</span></dt> <dd class="field-even"><dl class="simple"> <dt><strong>alphas</strong><span class="classifier">ndarray of shape (n_alphas,)</span></dt><dd><p>The alphas along the path where models are computed.</p> </dd> <dt><strong>coefs</strong><span class="classifier">ndarray of shape (n_features, n_alphas) or (n_targets, n_features, n_alphas)</span></dt><dd><p>Coefficients along the path.</p> </dd> <dt><strong>dual_gaps</strong><span class="classifier">ndarray of shape (n_alphas,)</span></dt><dd><p>The dual gaps at the end of the optimization for each alpha.</p> </dd> <dt><strong>n_iters</strong><span class="classifier">list of int</span></dt><dd><p>The number of iterations taken by the coordinate descent optimizer to reach the specified tolerance for each alpha.</p> </dd> </dl> </dd> </dl> <div class="admonition seealso"> <p class="admonition-title">See also</p> <dl class="simple"> <dt><a class="reference internal" href="sklearn.linear_model.lars_path.html#sklearn.linear_model.lars_path" title="sklearn.linear_model.lars_path"><code class="xref py py-obj docutils literal notranslate"><span class="pre">lars_path</span></code></a></dt><dd><p>Compute Least Angle Regression or Lasso path using LARS algorithm.</p> </dd> <dt><a class="reference internal" href="sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Lasso</span></code></a></dt><dd><p>The Lasso is a linear model that estimates sparse coefficients.</p> </dd> <dt><a class="reference internal" href="sklearn.linear_model.LassoLars.html#sklearn.linear_model.LassoLars" title="sklearn.linear_model.LassoLars"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LassoLars</span></code></a></dt><dd><p>Lasso model fit with Least Angle Regression a.k.a. Lars.</p> </dd> <dt><a class="reference internal" href="sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LassoCV</span></code></a></dt><dd><p>Lasso linear model with iterative fitting along a regularization path.</p> </dd> <dt><a class="reference internal" href="sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a></dt><dd><p>Cross-validated Lasso using the LARS algorithm.</p> </dd> <dt><a class="reference internal" href="sklearn.decomposition.sparse_encode.html#sklearn.decomposition.sparse_encode" title="sklearn.decomposition.sparse_encode"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sklearn.decomposition.sparse_encode</span></code></a></dt><dd><p>Estimator that can be used to transform signals into sparse linear combination of atoms from a fixed.</p> </dd> </dl> </div> <p class="rubric">Notes</p> <p>For an example, see <a class="reference internal" href="../../auto_examples/linear_model/plot_lasso_lasso_lars_elasticnet_path.html#sphx-glr-auto-examples-linear-model-plot-lasso-lasso-lars-elasticnet-path-py"><span class="std std-ref">examples/linear_model/plot_lasso_lasso_lars_elasticnet_path.py</span></a>.</p> <p>To avoid unnecessary memory duplication the X argument of the fit method should be directly passed as a Fortran-contiguous numpy array.</p> <p>Note that in certain cases, the Lars solver may be significantly faster to implement this functionality. In particular, linear interpolation can be used to retrieve model coefficients between the values output by lars_path</p> <p class="rubric">Examples</p> <p>Comparing lasso_path and lars_path with interpolation:</p> <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> <span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">lasso_path</span> <span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">3.1</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.3</span><span class="p">,</span> <span class="mf">5.4</span><span class="p">,</span> <span class="mf">4.3</span><span class="p">]])</span><span class="o">.</span><span class="n">T</span> <span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">3.1</span><span class="p">])</span> <span class="gp">>>> </span><span class="c1"># Use lasso_path to compute a coefficient path</span> <span class="gp">>>> </span><span class="n">_</span><span class="p">,</span> <span class="n">coef_path</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">lasso_path</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">alphas</span><span class="o">=</span><span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">])</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">coef_path</span><span class="p">)</span> <span class="go">[[0. 0. 0.46874778]</span> <span class="go"> [0.2159048 0.4425765 0.23689075]]</span> </pre></div> </div> <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># Now use lars_path and 1D linear interpolation to compute the</span> <span class="gp">>>> </span><span class="c1"># same path</span> <span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">lars_path</span> <span class="gp">>>> </span><span class="n">alphas</span><span class="p">,</span> <span class="n">active</span><span class="p">,</span> <span class="n">coef_path_lars</span> <span class="o">=</span> <span class="n">lars_path</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s1">'lasso'</span><span class="p">)</span> <span class="gp">>>> </span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">interpolate</span> <span class="gp">>>> </span><span class="n">coef_path_continuous</span> <span class="o">=</span> <span class="n">interpolate</span><span class="o">.</span><span class="n">interp1d</span><span class="p">(</span><span class="n">alphas</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="gp">... </span> <span class="n">coef_path_lars</span><span class="p">[:,</span> <span class="p">::</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">coef_path_continuous</span><span class="p">([</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]))</span> <span class="go">[[0. 0. 0.46915237]</span> <span class="go"> [0.2159048 0.4425765 0.23668876]]</span> </pre></div> </div> </dd></dl> <section id="gallery-examples"> <h2>Gallery examples<a class="headerlink" href="#gallery-examples" title="Link to this heading">#</a></h2> <div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example shows how to compute the "paths" of coefficients along the Lasso, Lasso-LARS, and Elastic Net regularization paths. In other words, it shows the relationship between the regularization parameter (alpha) and the coefficients."><img alt="" src="../../_images/sphx_glr_plot_lasso_lasso_lars_elasticnet_path_thumb.png" /> <p><a class="reference internal" href="../../auto_examples/linear_model/plot_lasso_lasso_lars_elasticnet_path.html#sphx-glr-auto-examples-linear-model-plot-lasso-lasso-lars-elasticnet-path-py"><span class="std std-ref">Lasso, Lasso-LARS, and Elastic Net paths</span></a></p> <div class="sphx-glr-thumbnail-title">Lasso, Lasso-LARS, and Elastic Net paths</div> </div></div></section> </section> </article> <footer class="bd-footer-article"> <div class="footer-article-items footer-article__inner"> <div class="footer-article-item"> <div class="prev-next-area"> <a class="left-prev" href="sklearn.linear_model.lars_path_gram.html" title="previous page"> <i class="fa-solid fa-angle-left"></i> <div class="prev-next-info"> <p class="prev-next-subtitle">previous</p> <p class="prev-next-title">lars_path_gram</p> </div> </a> <a class="right-next" href="sklearn.linear_model.orthogonal_mp.html" title="next page"> <div class="prev-next-info"> <p class="prev-next-subtitle">next</p> <p class="prev-next-title">orthogonal_mp</p> </div> <i class="fa-solid fa-angle-right"></i> </a> </div></div> </div> </footer> </div> <div class="bd-sidebar-secondary bd-toc"><div class="sidebar-secondary-items sidebar-secondary__inner"> <div class="sidebar-secondary-item"> <div id="pst-page-navigation-heading-2" class="page-toc tocsection onthispage"> <i class="fa-solid fa-list"></i> On this page </div> <nav class="bd-toc-nav page-toc" aria-labelledby="pst-page-navigation-heading-2"> <ul class="visible nav section-nav flex-column"> <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sklearn.linear_model.lasso_path"><code class="docutils literal notranslate"><span class="pre">lasso_path</span></code></a></li> <li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gallery-examples">Gallery examples</a></li> </ul> </nav></div> <div class="sidebar-secondary-item"> <div class="tocsection sourcelink"> <a href="../../_sources/modules/generated/sklearn.linear_model.lasso_path.rst.txt"> <i class="fa-solid fa-file-lines"></i> Show Source </a> </div> </div> </div></div> </div> <footer class="bd-footer-content"> </footer> </main> </div> </div> <!-- Scripts loaded after <body> so the DOM is not blocked --> <script src="../../_static/scripts/bootstrap.js?digest=dfe6caa3a7d634c4db9b"></script> <script src="../../_static/scripts/pydata-sphinx-theme.js?digest=dfe6caa3a7d634c4db9b"></script> <footer class="bd-footer"> <div class="bd-footer__inner bd-page-width"> <div class="footer-items__start"> <div class="footer-item"> <p class="copyright"> © Copyright 2007 - 2024, scikit-learn developers (BSD License). <br/> </p> </div> </div> </div> </footer> </body> </html>