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<section id="linear-models">
<span id="linear-model"></span><h1><span class="section-number">1.1. </span>Linear Models<a class="headerlink" href="#linear-models" title="Link to this heading">#</a></h1>
<p>The following are a set of methods intended for regression in which
the target value is expected to be a linear combination of the features.
In mathematical notation, if <span class="math notranslate nohighlight">\(\hat{y}\)</span> is the predicted
value.</p>
<div class="math notranslate nohighlight">
\[\hat{y}(w, x) = w_0 + w_1 x_1 + ... + w_p x_p\]</div>
<p>Across the module, we designate the vector <span class="math notranslate nohighlight">\(w = (w_1,
..., w_p)\)</span> as <code class="docutils literal notranslate"><span class="pre">coef_</span></code> and <span class="math notranslate nohighlight">\(w_0\)</span> as <code class="docutils literal notranslate"><span class="pre">intercept_</span></code>.</p>
<p>To perform classification with generalized linear models, see
<a class="reference internal" href="#logistic-regression"><span class="std std-ref">Logistic regression</span></a>.</p>
<section id="ordinary-least-squares">
<span id="id1"></span><h2><span class="section-number">1.1.1. </span>Ordinary Least Squares<a class="headerlink" href="#ordinary-least-squares" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearRegression</span></code></a> fits a linear model with coefficients
<span class="math notranslate nohighlight">\(w = (w_1, ..., w_p)\)</span> to minimize the residual sum
of squares between the observed targets in the dataset, and the
targets predicted by the linear approximation. Mathematically it
solves a problem of the form:</p>
<div class="math notranslate nohighlight">
\[\min_{w} || X w - y||_2^2\]</div>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_ols_ridge.html"><img alt="../_images/sphx_glr_plot_ols_ridge_001.png" src="../_images/sphx_glr_plot_ols_ridge_001.png" style="width: 500.0px; height: 250.0px;" />
</a>
</figure>
<p><a class="reference internal" href="generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearRegression</span></code></a> takes in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method arguments <code class="docutils literal notranslate"><span class="pre">X</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code>,
<code class="docutils literal notranslate"><span class="pre">sample_weight</span></code> and stores the coefficients <span class="math notranslate nohighlight">\(w\)</span> of the linear model in its
<code class="docutils literal notranslate"><span class="pre">coef_</span></code> and <code class="docutils literal notranslate"><span class="pre">intercept_</span></code> attributes:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</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="go">LinearRegression()</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.5, 0.5])</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">intercept_</span>
<span class="go">0.0</span>
</pre></div>
</div>
<p>The coefficient estimates for Ordinary Least Squares rely on the
independence of the features. When features are correlated and some
columns of the design matrix <span class="math notranslate nohighlight">\(X\)</span> have an approximately linear
dependence, the design matrix becomes close to singular
and as a result, the least-squares estimate becomes highly sensitive
to random errors in the observed target, producing a large
variance. This situation of <em>multicollinearity</em> can arise, for
example, when data are collected without an experimental design.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ols_ridge.html#sphx-glr-auto-examples-linear-model-plot-ols-ridge-py"><span class="std std-ref">Ordinary Least Squares and Ridge Regression</span></a></p></li>
</ul>
<section id="non-negative-least-squares">
<h3><span class="section-number">1.1.1.1. </span>Non-Negative Least Squares<a class="headerlink" href="#non-negative-least-squares" title="Link to this heading">#</a></h3>
<p>It is possible to constrain all the coefficients to be non-negative, which may
be useful when they represent some physical or naturally non-negative
quantities (e.g., frequency counts or prices of goods).
<a class="reference internal" href="generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearRegression</span></code></a> accepts a boolean <code class="docutils literal notranslate"><span class="pre">positive</span></code>
parameter: when set to <code class="docutils literal notranslate"><span class="pre">True</span></code> <a class="reference external" href="https://en.wikipedia.org/wiki/Non-negative_least_squares">Non-Negative Least Squares</a> are then applied.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_nnls.html#sphx-glr-auto-examples-linear-model-plot-nnls-py"><span class="std std-ref">Non-negative least squares</span></a></p></li>
</ul>
</section>
<section id="ordinary-least-squares-complexity">
<h3><span class="section-number">1.1.1.2. </span>Ordinary Least Squares Complexity<a class="headerlink" href="#ordinary-least-squares-complexity" title="Link to this heading">#</a></h3>
<p>The least squares solution is computed using the singular value
decomposition of <span class="math notranslate nohighlight">\(X\)</span>. If <span class="math notranslate nohighlight">\(X\)</span> is a matrix of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_features)</span></code>
this method has a cost of
<span class="math notranslate nohighlight">\(O(n_{\text{samples}} n_{\text{features}}^2)\)</span>, assuming that
<span class="math notranslate nohighlight">\(n_{\text{samples}} \geq n_{\text{features}}\)</span>.</p>
</section>
</section>
<section id="ridge-regression-and-classification">
<span id="ridge-regression"></span><h2><span class="section-number">1.1.2. </span>Ridge regression and classification<a class="headerlink" href="#ridge-regression-and-classification" title="Link to this heading">#</a></h2>
<section id="regression">
<h3><span class="section-number">1.1.2.1. </span>Regression<a class="headerlink" href="#regression" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> regression addresses some of the problems of
<a class="reference internal" href="#ordinary-least-squares"><span class="std std-ref">Ordinary Least Squares</span></a> by imposing a penalty on the size of the
coefficients. The ridge coefficients minimize a penalized residual sum
of squares:</p>
<div class="math notranslate nohighlight">
\[\min_{w} || X w - y||_2^2 + \alpha ||w||_2^2\]</div>
<p>The complexity parameter <span class="math notranslate nohighlight">\(\alpha \geq 0\)</span> controls the amount
of shrinkage: the larger the value of <span class="math notranslate nohighlight">\(\alpha\)</span>, the greater the amount
of shrinkage and thus the coefficients become more robust to collinearity.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_ridge_path.html"><img alt="../_images/sphx_glr_plot_ridge_path_001.png" src="../_images/sphx_glr_plot_ridge_path_001.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<p>As with other linear models, <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> will take in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method
arrays <code class="docutils literal notranslate"><span class="pre">X</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code> and will store the coefficients <span class="math notranslate nohighlight">\(w\)</span> of the linear model in
its <code class="docutils literal notranslate"><span class="pre">coef_</span></code> member:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">Ridge(alpha=0.5)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.34545455, 0.34545455])</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">intercept_</span>
<span class="go">np.float64(0.13636...)</span>
</pre></div>
</div>
<p>Note that the class <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> allows for the user to specify that the
solver be automatically chosen by setting <code class="docutils literal notranslate"><span class="pre">solver="auto"</span></code>. When this option
is specified, <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> will choose between the <code class="docutils literal notranslate"><span class="pre">"lbfgs"</span></code>, <code class="docutils literal notranslate"><span class="pre">"cholesky"</span></code>,
and <code class="docutils literal notranslate"><span class="pre">"sparse_cg"</span></code> solvers. <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> will begin checking the conditions
shown in the following table from top to bottom. If the condition is true,
the corresponding solver is chosen.</p>
<div class="pst-scrollable-table-container"><table class="table">
<tbody>
<tr class="row-odd"><td><p><strong>Solver</strong></p></td>
<td><p><strong>Condition</strong></p></td>
</tr>
<tr class="row-even"><td><p>‘lbfgs’</p></td>
<td><p>The <code class="docutils literal notranslate"><span class="pre">positive=True</span></code> option is specified.</p></td>
</tr>
<tr class="row-odd"><td><p>‘cholesky’</p></td>
<td><p>The input array X is not sparse.</p></td>
</tr>
<tr class="row-even"><td><p>‘sparse_cg’</p></td>
<td><p>None of the above conditions are fulfilled.</p></td>
</tr>
</tbody>
</table>
</div>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ols_ridge.html#sphx-glr-auto-examples-linear-model-plot-ols-ridge-py"><span class="std std-ref">Ordinary Least Squares and Ridge Regression</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ridge_path.html#sphx-glr-auto-examples-linear-model-plot-ridge-path-py"><span class="std std-ref">Plot Ridge coefficients as a function of the regularization</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_linear_model_coefficient_interpretation.html#sphx-glr-auto-examples-inspection-plot-linear-model-coefficient-interpretation-py"><span class="std std-ref">Common pitfalls in the interpretation of coefficients of linear models</span></a></p></li>
</ul>
</section>
<section id="classification">
<h3><span class="section-number">1.1.2.2. </span>Classification<a class="headerlink" href="#classification" title="Link to this heading">#</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> regressor has a classifier variant:
<a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifier</span></code></a>. This classifier first converts binary targets to
<code class="docutils literal notranslate"><span class="pre">{-1,</span> <span class="pre">1}</span></code> and then treats the problem as a regression task, optimizing the
same objective as above. The predicted class corresponds to the sign of the
regressor’s prediction. For multiclass classification, the problem is
treated as multi-output regression, and the predicted class corresponds to
the output with the highest value.</p>
<p>It might seem questionable to use a (penalized) Least Squares loss to fit a
classification model instead of the more traditional logistic or hinge
losses. However, in practice, all those models can lead to similar
cross-validation scores in terms of accuracy or precision/recall, while the
penalized least squares loss used by the <a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifier</span></code></a> allows for
a very different choice of the numerical solvers with distinct computational
performance profiles.</p>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifier</span></code></a> can be significantly faster than e.g.
<a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a> with a high number of classes because it can
compute the projection matrix <span class="math notranslate nohighlight">\((X^T X)^{-1} X^T\)</span> only once.</p>
<p>This classifier is sometimes referred to as a <a class="reference external" href="https://en.wikipedia.org/wiki/Least-squares_support-vector_machine">Least Squares Support Vector
Machine</a> with
a linear kernel.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/text/plot_document_classification_20newsgroups.html#sphx-glr-auto-examples-text-plot-document-classification-20newsgroups-py"><span class="std std-ref">Classification of text documents using sparse features</span></a></p></li>
</ul>
</section>
<section id="ridge-complexity">
<h3><span class="section-number">1.1.2.3. </span>Ridge Complexity<a class="headerlink" href="#ridge-complexity" title="Link to this heading">#</a></h3>
<p>This method has the same order of complexity as
<a class="reference internal" href="#ordinary-least-squares"><span class="std std-ref">Ordinary Least Squares</span></a>.</p>
</section>
<section id="setting-the-regularization-parameter-leave-one-out-cross-validation">
<h3><span class="section-number">1.1.2.4. </span>Setting the regularization parameter: leave-one-out Cross-Validation<a class="headerlink" href="#setting-the-regularization-parameter-leave-one-out-cross-validation" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="sklearn.linear_model.RidgeCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeCV</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifierCV.html#sklearn.linear_model.RidgeClassifierCV" title="sklearn.linear_model.RidgeClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifierCV</span></code></a> implement ridge
regression/classification with built-in cross-validation of the alpha parameter.
They work in the same way as <a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridSearchCV</span></code></a> except
that it defaults to efficient Leave-One-Out <a class="reference internal" href="../glossary.html#term-cross-validation"><span class="xref std std-term">cross-validation</span></a>.
When using the default <a class="reference internal" href="../glossary.html#term-cross-validation"><span class="xref std std-term">cross-validation</span></a>, alpha cannot be 0 due to the
formulation used to calculate Leave-One-Out error. See <a class="reference internal" href="#rl2007" id="id3"><span>[RL2007]</span></a> for details.</p>
<p>Usage example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">RidgeCV</span><span class="p">(</span><span class="n">alphas</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">13</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">RidgeCV(alphas=array([1.e-06, 1.e-05, 1.e-04, 1.e-03, 1.e-02, 1.e-01, 1.e+00, 1.e+01,</span>
<span class="go"> 1.e+02, 1.e+03, 1.e+04, 1.e+05, 1.e+06]))</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">alpha_</span>
<span class="go">np.float64(0.01)</span>
</pre></div>
</div>
<p>Specifying the value of the <a class="reference internal" href="../glossary.html#term-cv"><span class="xref std std-term">cv</span></a> attribute will trigger the use of
cross-validation with <a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridSearchCV</span></code></a>, for
example <code class="docutils literal notranslate"><span class="pre">cv=10</span></code> for 10-fold cross-validation, rather than Leave-One-Out
Cross-Validation.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<div role="list" class="citation-list">
<div class="citation" id="rl2007" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">RL2007</a><span class="fn-bracket">]</span></span>
<p class="sd-card-text">“Notes on Regularized Least Squares”, Rifkin & Lippert (<a class="reference external" href="http://cbcl.mit.edu/publications/ps/MIT-CSAIL-TR-2007-025.pdf">technical report</a>,
<a class="reference external" href="https://www.mit.edu/~9.520/spring07/Classes/rlsslides.pdf">course slides</a>).</p>
</div>
</div>
</div>
</details></section>
</section>
<section id="lasso">
<span id="id4"></span><h2><span class="section-number">1.1.3. </span>Lasso<a class="headerlink" href="#lasso" title="Link to this heading">#</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">Lasso</span></code></a> is a linear model that estimates sparse coefficients.
It is useful in some contexts due to its tendency to prefer solutions
with fewer non-zero coefficients, effectively reducing the number of
features upon which the given solution is dependent. For this reason,
Lasso and its variants are fundamental to the field of compressed sensing.
Under certain conditions, it can recover the exact set of non-zero
coefficients (see
<a class="reference internal" href="../auto_examples/applications/plot_tomography_l1_reconstruction.html#sphx-glr-auto-examples-applications-plot-tomography-l1-reconstruction-py"><span class="std std-ref">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</span></a>).</p>
<p>Mathematically, it consists of a linear model with an added regularization term.
The objective function to minimize is:</p>
<div class="math notranslate nohighlight">
\[\min_{w} { \frac{1}{2n_{\text{samples}}} ||X w - y||_2 ^ 2 + \alpha ||w||_1}\]</div>
<p>The lasso estimate thus solves the minimization of the
least-squares penalty with <span class="math notranslate nohighlight">\(\alpha ||w||_1\)</span> added, where
<span class="math notranslate nohighlight">\(\alpha\)</span> is a constant and <span class="math notranslate nohighlight">\(||w||_1\)</span> is the <span class="math notranslate nohighlight">\(\ell_1\)</span>-norm of
the coefficient vector.</p>
<p>The implementation in the class <a class="reference internal" href="generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">Lasso</span></code></a> uses coordinate descent as
the algorithm to fit the coefficients. See <a class="reference internal" href="#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a>
for another implementation:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Lasso</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">Lasso(alpha=0.1)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="go">array([0.8])</span>
</pre></div>
</div>
<p>The function <a class="reference internal" href="generated/sklearn.linear_model.lasso_path.html#sklearn.linear_model.lasso_path" title="sklearn.linear_model.lasso_path"><code class="xref py py-func docutils literal notranslate"><span class="pre">lasso_path</span></code></a> is useful for lower-level tasks, as it
computes the coefficients along the full path of possible values.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_and_elasticnet.html#sphx-glr-auto-examples-linear-model-plot-lasso-and-elasticnet-py"><span class="std std-ref">L1-based models for Sparse Signals</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/applications/plot_tomography_l1_reconstruction.html#sphx-glr-auto-examples-applications-plot-tomography-l1-reconstruction-py"><span class="std std-ref">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_linear_model_coefficient_interpretation.html#sphx-glr-auto-examples-inspection-plot-linear-model-coefficient-interpretation-py"><span class="std std-ref">Common pitfalls in the interpretation of coefficients of linear models</span></a></p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><strong>Feature selection with Lasso</strong></p>
<p>As the Lasso regression yields sparse models, it can
thus be used to perform feature selection, as detailed in
<a class="reference internal" href="feature_selection.html#l1-feature-selection"><span class="std std-ref">L1-based feature selection</span></a>.</p>
</div>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references-2">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references-2" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The following two references explain the iterations
used in the coordinate descent solver of scikit-learn, as well as
the duality gap computation used for convergence control.</p>
<ul class="simple">
<li><p class="sd-card-text">“Regularization Path For Generalized linear Models by Coordinate Descent”,
Friedman, Hastie & Tibshirani, J Stat Softw, 2010 (<a class="reference external" href="https://www.jstatsoft.org/article/view/v033i01/v33i01.pdf">Paper</a>).</p></li>
<li><p class="sd-card-text">“An Interior-Point Method for Large-Scale L1-Regularized Least Squares,”
S. J. Kim, K. Koh, M. Lustig, S. Boyd and D. Gorinevsky,
in IEEE Journal of Selected Topics in Signal Processing, 2007
(<a class="reference external" href="https://web.stanford.edu/~boyd/papers/pdf/l1_ls.pdf">Paper</a>)</p></li>
</ul>
</div>
</details><section id="setting-regularization-parameter">
<h3><span class="section-number">1.1.3.1. </span>Setting regularization parameter<a class="headerlink" href="#setting-regularization-parameter" title="Link to this heading">#</a></h3>
<p>The <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter controls the degree of sparsity of the estimated
coefficients.</p>
<section id="using-cross-validation">
<h4><span class="section-number">1.1.3.1.1. </span>Using cross-validation<a class="headerlink" href="#using-cross-validation" title="Link to this heading">#</a></h4>
<p>scikit-learn exposes objects that set the Lasso <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter by
cross-validation: <a class="reference internal" href="generated/sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoCV</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a>.
<a class="reference internal" href="generated/sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a> is based on the <a class="reference internal" href="#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a> algorithm
explained below.</p>
<p>For high-dimensional datasets with many collinear features,
<a class="reference internal" href="generated/sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoCV</span></code></a> is most often preferable. However, <a class="reference internal" href="generated/sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a> has
the advantage of exploring more relevant values of <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter, and
if the number of samples is very small compared to the number of
features, it is often faster than <a class="reference internal" href="generated/sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoCV</span></code></a>.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/linear_model/plot_lasso_model_selection.html"><img alt="lasso_cv_1" src="../_images/sphx_glr_plot_lasso_model_selection_002.png" style="width: 307.2px; height: 230.39999999999998px;" /></a> <a class="reference external" href="../auto_examples/linear_model/plot_lasso_model_selection.html"><img alt="lasso_cv_2" src="../_images/sphx_glr_plot_lasso_model_selection_003.png" style="width: 307.2px; height: 230.39999999999998px;" /></a></strong></p></section>
<section id="information-criteria-based-model-selection">
<span id="lasso-lars-ic"></span><h4><span class="section-number">1.1.3.1.2. </span>Information-criteria based model selection<a class="headerlink" href="#information-criteria-based-model-selection" title="Link to this heading">#</a></h4>
<p>Alternatively, the estimator <a class="reference internal" href="generated/sklearn.linear_model.LassoLarsIC.html#sklearn.linear_model.LassoLarsIC" title="sklearn.linear_model.LassoLarsIC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsIC</span></code></a> proposes to use the
Akaike information criterion (AIC) and the Bayes Information criterion (BIC).
It is a computationally cheaper alternative to find the optimal value of alpha
as the regularization path is computed only once instead of k+1 times
when using k-fold cross-validation.</p>
<p>Indeed, these criteria are computed on the in-sample training set. In short,
they penalize the over-optimistic scores of the different Lasso models by
their flexibility (cf. to “Mathematical details” section below).</p>
<p>However, such criteria need a proper estimation of the degrees of freedom of
the solution, are derived for large samples (asymptotic results) and assume the
correct model is candidates under investigation. They also tend to break when
the problem is badly conditioned (e.g. more features than samples).</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_lasso_lars_ic.html"><img alt="../_images/sphx_glr_plot_lasso_lars_ic_001.png" src="../_images/sphx_glr_plot_lasso_lars_ic_001.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_model_selection.html#sphx-glr-auto-examples-linear-model-plot-lasso-model-selection-py"><span class="std std-ref">Lasso model selection: AIC-BIC / cross-validation</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_lars_ic.html#sphx-glr-auto-examples-linear-model-plot-lasso-lars-ic-py"><span class="std std-ref">Lasso model selection via information criteria</span></a></p></li>
</ul>
</section>
<section id="aic-and-bic-criteria">
<span id="aic-bic"></span><h4><span class="section-number">1.1.3.1.3. </span>AIC and BIC criteria<a class="headerlink" href="#aic-and-bic-criteria" title="Link to this heading">#</a></h4>
<p>The definition of AIC (and thus BIC) might differ in the literature. In this
section, we give more information regarding the criterion computed in
scikit-learn.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="mathematical-details">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Mathematical details<a class="headerlink" href="#mathematical-details" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The AIC criterion is defined as:</p>
<div class="math notranslate nohighlight">