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<ul>
<li><a class="reference internal" href="#">1.1. Linear Models</a><ul>
<li><a class="reference internal" href="#ordinary-least-squares">1.1.1. Ordinary Least Squares</a><ul>
<li><a class="reference internal" href="#non-negative-least-squares">1.1.1.1. Non-Negative Least Squares</a></li>
<li><a class="reference internal" href="#ordinary-least-squares-complexity">1.1.1.2. Ordinary Least Squares Complexity</a></li>
</ul>
</li>
<li><a class="reference internal" href="#ridge-regression-and-classification">1.1.2. Ridge regression and classification</a><ul>
<li><a class="reference internal" href="#regression">1.1.2.1. Regression</a></li>
<li><a class="reference internal" href="#classification">1.1.2.2. Classification</a></li>
<li><a class="reference internal" href="#ridge-complexity">1.1.2.3. Ridge Complexity</a></li>
<li><a class="reference internal" href="#setting-the-regularization-parameter-leave-one-out-cross-validation">1.1.2.4. Setting the regularization parameter: leave-one-out Cross-Validation</a></li>
</ul>
</li>
<li><a class="reference internal" href="#lasso">1.1.3. Lasso</a><ul>
<li><a class="reference internal" href="#setting-regularization-parameter">1.1.3.1. Setting regularization parameter</a><ul>
<li><a class="reference internal" href="#using-cross-validation">1.1.3.1.1. Using cross-validation</a></li>
<li><a class="reference internal" href="#information-criteria-based-model-selection">1.1.3.1.2. Information-criteria based model selection</a></li>
<li><a class="reference internal" href="#comparison-with-the-regularization-parameter-of-svm">1.1.3.1.3. Comparison with the regularization parameter of SVM</a></li>
</ul>
</li>
</ul>
</li>
<li><a class="reference internal" href="#multi-task-lasso">1.1.4. Multi-task Lasso</a></li>
<li><a class="reference internal" href="#elastic-net">1.1.5. Elastic-Net</a></li>
<li><a class="reference internal" href="#multi-task-elastic-net">1.1.6. Multi-task Elastic-Net</a></li>
<li><a class="reference internal" href="#least-angle-regression">1.1.7. Least Angle Regression</a></li>
<li><a class="reference internal" href="#lars-lasso">1.1.8. LARS Lasso</a><ul>
<li><a class="reference internal" href="#mathematical-formulation">1.1.8.1. Mathematical formulation</a></li>
</ul>
</li>
<li><a class="reference internal" href="#orthogonal-matching-pursuit-omp">1.1.9. Orthogonal Matching Pursuit (OMP)</a></li>
<li><a class="reference internal" href="#bayesian-regression">1.1.10. Bayesian Regression</a><ul>
<li><a class="reference internal" href="#bayesian-ridge-regression">1.1.10.1. Bayesian Ridge Regression</a></li>
<li><a class="reference internal" href="#automatic-relevance-determination-ard">1.1.10.2. Automatic Relevance Determination - ARD</a></li>
</ul>
</li>
<li><a class="reference internal" href="#logistic-regression">1.1.11. Logistic regression</a><ul>
<li><a class="reference internal" href="#binary-case">1.1.11.1. Binary Case</a></li>
<li><a class="reference internal" href="#multinomial-case">1.1.11.2. Multinomial Case</a></li>
<li><a class="reference internal" href="#solvers">1.1.11.3. Solvers</a></li>
</ul>
</li>
<li><a class="reference internal" href="#generalized-linear-regression">1.1.12. Generalized Linear Regression</a><ul>
<li><a class="reference internal" href="#usage">1.1.12.1. Usage</a></li>
<li><a class="reference internal" href="#practical-considerations">1.1.12.2. Practical considerations</a></li>
</ul>
</li>
<li><a class="reference internal" href="#stochastic-gradient-descent-sgd">1.1.13. Stochastic Gradient Descent - SGD</a></li>
<li><a class="reference internal" href="#perceptron">1.1.14. Perceptron</a></li>
<li><a class="reference internal" href="#passive-aggressive-algorithms">1.1.15. Passive Aggressive Algorithms</a></li>
<li><a class="reference internal" href="#robustness-regression-outliers-and-modeling-errors">1.1.16. Robustness regression: outliers and modeling errors</a><ul>
<li><a class="reference internal" href="#different-scenario-and-useful-concepts">1.1.16.1. Different scenario and useful concepts</a></li>
<li><a class="reference internal" href="#ransac-random-sample-consensus">1.1.16.2. RANSAC: RANdom SAmple Consensus</a><ul>
<li><a class="reference internal" href="#details-of-the-algorithm">1.1.16.2.1. Details of the algorithm</a></li>
</ul>
</li>
<li><a class="reference internal" href="#theil-sen-estimator-generalized-median-based-estimator">1.1.16.3. Theil-Sen estimator: generalized-median-based estimator</a><ul>
<li><a class="reference internal" href="#theoretical-considerations">1.1.16.3.1. Theoretical considerations</a></li>
</ul>
</li>
<li><a class="reference internal" href="#huber-regression">1.1.16.4. Huber Regression</a></li>
<li><a class="reference internal" href="#notes">1.1.16.5. Notes</a></li>
</ul>
</li>
<li><a class="reference internal" href="#quantile-regression">1.1.17. Quantile Regression</a></li>
<li><a class="reference internal" href="#polynomial-regression-extending-linear-models-with-basis-functions">1.1.18. Polynomial regression: extending linear models with basis functions</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<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="Permalink 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="Permalink 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.html"><img alt="../_images/sphx_glr_plot_ols_001.png" src="../_images/sphx_glr_plot_ols_001.png" style="width: 320.0px; height: 240.0px;" /></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> will take in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method arrays X, y
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="nn">sklearn</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>
</pre></div>
</div>
<p>The coefficient estimates for Ordinary Least Squares rely on the
independence of the features. When features are correlated and the
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>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ols.html#sphx-glr-auto-examples-linear-model-plot-ols-py"><span class="std std-ref">Linear Regression Example</span></a></p></li>
</ul>
</aside>
<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="Permalink 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>
<aside class="topic">
<p class="topic-title">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>
</aside>
</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="Permalink to this heading">¶</a></h3>
<p>The least squares solution is computed using the singular value
decomposition of X. If X 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="Permalink 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="Permalink 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 X, y 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="nn">sklearn</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">0.13636...</span>
</pre></div>
</div>
</section>
<section id="classification">
<h3><span class="section-number">1.1.2.2. </span>Classification<a class="headerlink" href="#classification" title="Permalink 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
Machines</a> with
a linear kernel.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<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/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>
<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>
</aside>
</section>
<section id="ridge-complexity">
<h3><span class="section-number">1.1.2.3. </span>Ridge Complexity<a class="headerlink" href="#ridge-complexity" title="Permalink 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="Permalink 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> implements ridge regression with built-in
cross-validation of the alpha parameter. The object works in the same way
as GridSearchCV except that it defaults to Leave-One-Out Cross-Validation:</p>
<div class="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</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">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>
<aside class="topic">
<p class="topic-title">References</p>
<ul class="simple">
<li><p>“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></li>
</ul>
</aside>
</section>
</section>
<section id="lasso">
<span id="id3"></span><h2><span class="section-number">1.1.3. </span>Lasso<a class="headerlink" href="#lasso" title="Permalink 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="nn">sklearn</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>
<aside class="topic">
<p class="topic-title">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">Lasso and Elastic Net 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>
</aside>
<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>
<p>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>
<aside class="topic">
<p class="topic-title">References</p>
<ul class="simple">
<li><p>“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>“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>
</aside>
<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="Permalink 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="Permalink 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="Permalink 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 id="aic-bic"><strong>Mathematical details</strong></p>
<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. The AIC criterion is defined as:</p>
<div class="math notranslate nohighlight">
\[AIC = -2 \log(\hat{L}) + 2 d\]</div>
<p>where <span class="math notranslate nohighlight">\(\hat{L}\)</span> is the maximum likelihood of the model and
<span class="math notranslate nohighlight">\(d\)</span> is the number of parameters (as well referred to as degrees of
freedom in the previous section).</p>
<p>The definition of BIC replace the constant <span class="math notranslate nohighlight">\(2\)</span> by <span class="math notranslate nohighlight">\(\log(N)\)</span>:</p>
<div class="math notranslate nohighlight">
\[BIC = -2 \log(\hat{L}) + \log(N) d\]</div>
<p>where <span class="math notranslate nohighlight">\(N\)</span> is the number of samples.</p>
<p>For a linear Gaussian model, the maximum log-likelihood is defined as:</p>
<div class="math notranslate nohighlight">
\[\log(\hat{L}) = - \frac{n}{2} \log(2 \pi) - \frac{n}{2} \ln(\sigma^2) - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{2\sigma^2}\]</div>
<p>where <span class="math notranslate nohighlight">\(\sigma^2\)</span> is an estimate of the noise variance,
<span class="math notranslate nohighlight">\(y_i\)</span> and <span class="math notranslate nohighlight">\(\hat{y}_i\)</span> are respectively the true and predicted
targets, and <span class="math notranslate nohighlight">\(n\)</span> is the number of samples.</p>
<p>Plugging the maximum log-likelihood in the AIC formula yields:</p>
<div class="math notranslate nohighlight">
\[AIC = n \log(2 \pi \sigma^2) + \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sigma^2} + 2 d\]</div>
<p>The first term of the above expression is sometimes discarded since it is a
constant when <span class="math notranslate nohighlight">\(\sigma^2\)</span> is provided. In addition,
it is sometimes stated that the AIC is equivalent to the <span class="math notranslate nohighlight">\(C_p\)</span> statistic
<a class="footnote-reference brackets" href="#id6" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>12<span class="fn-bracket">]</span></a>. In a strict sense, however, it is equivalent only up to some constant
and a multiplicative factor.</p>
<p>At last, we mentioned above that <span class="math notranslate nohighlight">\(\sigma^2\)</span> is an estimate of the
noise variance. In <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> when the parameter <code class="docutils literal notranslate"><span class="pre">noise_variance</span></code> is
not provided (default), the noise variance is estimated via the unbiased
estimator <a class="footnote-reference brackets" href="#id7" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>13<span class="fn-bracket">]</span></a> defined as:</p>
<div class="math notranslate nohighlight">
\[\sigma^2 = \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{n - p}\]</div>
<p>where <span class="math notranslate nohighlight">\(p\)</span> is the number of features and <span class="math notranslate nohighlight">\(\hat{y}_i\)</span> is the
predicted target using an ordinary least squares regression. Note, that this
formula is valid only when <code class="docutils literal notranslate"><span class="pre">n_samples</span> <span class="pre">></span> <span class="pre">n_features</span></code>.</p>
<aside class="topic">
<p class="topic-title">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>
</aside>
<aside class="topic">
<p class="topic-title">References</p>
<aside class="footnote brackets" id="id6" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">12</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://arxiv.org/abs/0712.0881.pdf">Zou, Hui, Trevor Hastie, and Robert Tibshirani.
“On the degrees of freedom of the lasso.”
The Annals of Statistics 35.5 (2007): 2173-2192.</a></p>
</aside>
<aside class="footnote brackets" id="id7" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">13</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1162/089976603321891864">Cherkassky, Vladimir, and Yunqian Ma.
“Comparison of model selection for regression.”
Neural computation 15.7 (2003): 1691-1714.</a></p>
</aside>
</aside>
</section>
<section id="comparison-with-the-regularization-parameter-of-svm">
<h4><span class="section-number">1.1.3.1.3. </span>Comparison with the regularization parameter of SVM<a class="headerlink" href="#comparison-with-the-regularization-parameter-of-svm" title="Permalink to this heading">¶</a></h4>
<p>The equivalence between <code class="docutils literal notranslate"><span class="pre">alpha</span></code> and the regularization parameter of SVM,
<code class="docutils literal notranslate"><span class="pre">C</span></code> is given by <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">=</span> <span class="pre">1</span> <span class="pre">/</span> <span class="pre">C</span></code> or <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">=</span> <span class="pre">1</span> <span class="pre">/</span> <span class="pre">(n_samples</span> <span class="pre">*</span> <span class="pre">C)</span></code>,
depending on the estimator and the exact objective function optimized by the
model.</p>
</section>
</section>
</section>
<section id="multi-task-lasso">
<span id="id8"></span><h2><span class="section-number">1.1.4. </span>Multi-task Lasso<a class="headerlink" href="#multi-task-lasso" title="Permalink to this heading">¶</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.MultiTaskLasso.html#sklearn.linear_model.MultiTaskLasso" title="sklearn.linear_model.MultiTaskLasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">MultiTaskLasso</span></code></a> is a linear model that estimates sparse
coefficients for multiple regression problems jointly: <code class="docutils literal notranslate"><span class="pre">y</span></code> is a 2D array,
of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_tasks)</span></code>. The constraint is that the selected
features are the same for all the regression problems, also called tasks.</p>
<p>The following figure compares the location of the non-zero entries in the
coefficient matrix W obtained with a simple Lasso or a MultiTaskLasso.
The Lasso estimates yield scattered non-zeros while the non-zeros of
the MultiTaskLasso are full columns.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/linear_model/plot_multi_task_lasso_support.html"><img alt="multi_task_lasso_1" src="../_images/sphx_glr_plot_multi_task_lasso_support_001.png" style="width: 384.0px; height: 240.0px;" /></a> <a class="reference external" href="../auto_examples/linear_model/plot_multi_task_lasso_support.html"><img alt="multi_task_lasso_2" src="../_images/sphx_glr_plot_multi_task_lasso_support_002.png" style="width: 307.2px; height: 230.39999999999998px;" /></a></strong></p><p class="centered">
<strong>Fitting a time-series model, imposing that any active feature be active at all times.</strong></p><aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_multi_task_lasso_support.html#sphx-glr-auto-examples-linear-model-plot-multi-task-lasso-support-py"><span class="std std-ref">Joint feature selection with multi-task Lasso</span></a></p></li>
</ul>
</aside>
<p>Mathematically, it consists of a linear model trained with a mixed
<span class="math notranslate nohighlight">\(\ell_1\)</span> <span class="math notranslate nohighlight">\(\ell_2\)</span>-norm for regularization.
The objective function to minimize is:</p>
<div class="math notranslate nohighlight">
\[\min_{W} { \frac{1}{2n_{\text{samples}}} ||X W - Y||_{\text{Fro}} ^ 2 + \alpha ||W||_{21}}\]</div>
<p>where <span class="math notranslate nohighlight">\(\text{Fro}\)</span> indicates the Frobenius norm</p>
<div class="math notranslate nohighlight">
\[||A||_{\text{Fro}} = \sqrt{\sum_{ij} a_{ij}^2}\]</div>
<p>and <span class="math notranslate nohighlight">\(\ell_1\)</span> <span class="math notranslate nohighlight">\(\ell_2\)</span> reads</p>
<div class="math notranslate nohighlight">
\[||A||_{2 1} = \sum_i \sqrt{\sum_j a_{ij}^2}.\]</div>
<p>The implementation in the class <a class="reference internal" href="generated/sklearn.linear_model.MultiTaskLasso.html#sklearn.linear_model.MultiTaskLasso" title="sklearn.linear_model.MultiTaskLasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">MultiTaskLasso</span></code></a> uses
coordinate descent as the algorithm to fit the coefficients.</p>
</section>
<section id="elastic-net">
<span id="id9"></span><h2><span class="section-number">1.1.5. </span>Elastic-Net<a class="headerlink" href="#elastic-net" title="Permalink to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.linear_model.ElasticNet.html#sklearn.linear_model.ElasticNet" title="sklearn.linear_model.ElasticNet"><code class="xref py py-class docutils literal notranslate"><span class="pre">ElasticNet</span></code></a> is a linear regression model trained with both
<span class="math notranslate nohighlight">\(\ell_1\)</span> and <span class="math notranslate nohighlight">\(\ell_2\)</span>-norm regularization of the coefficients.
This combination allows for learning a sparse model where few of
the weights are non-zero like <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>, while still maintaining
the regularization properties of <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>. We control the convex
combination of <span class="math notranslate nohighlight">\(\ell_1\)</span> and <span class="math notranslate nohighlight">\(\ell_2\)</span> using the <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code>
parameter.</p>
<p>Elastic-net is useful when there are multiple features that are
correlated with one another. Lasso is likely to pick one of these
at random, while elastic-net is likely to pick both.</p>
<p>A practical advantage of trading-off between Lasso and Ridge is that it
allows Elastic-Net to inherit some of Ridge’s stability under rotation.</p>
<p>The objective function to minimize is in this case</p>
<div class="math notranslate nohighlight">
\[\min_{w} { \frac{1}{2n_{\text{samples}}} ||X w - y||_2 ^ 2 + \alpha \rho ||w||_1 +
\frac{\alpha(1-\rho)}{2} ||w||_2 ^ 2}\]</div>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_lasso_coordinate_descent_path.html"><img alt="../_images/sphx_glr_plot_lasso_coordinate_descent_path_001.png" src="../_images/sphx_glr_plot_lasso_coordinate_descent_path_001.png" style="width: 320.0px; height: 240.0px;" /></a>
</figure>
<p>The class <a class="reference internal" href="generated/sklearn.linear_model.ElasticNetCV.html#sklearn.linear_model.ElasticNetCV" title="sklearn.linear_model.ElasticNetCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">ElasticNetCV</span></code></a> can be used to set the parameters
<code class="docutils literal notranslate"><span class="pre">alpha</span></code> (<span class="math notranslate nohighlight">\(\alpha\)</span>) and <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code> (<span class="math notranslate nohighlight">\(\rho\)</span>) by cross-validation.</p>
<aside class="topic">
<p class="topic-title">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">Lasso and Elastic Net for Sparse Signals</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_coordinate_descent_path.html#sphx-glr-auto-examples-linear-model-plot-lasso-coordinate-descent-path-py"><span class="std std-ref">Lasso and Elastic Net</span></a></p></li>
</ul>
</aside>
<p>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>
<aside class="topic">
<p class="topic-title">References</p>
<ul class="simple">
<li><p>“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>“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>
</aside>
</section>
<section id="multi-task-elastic-net">
<span id="id10"></span><h2><span class="section-number">1.1.6. </span>Multi-task Elastic-Net<a class="headerlink" href="#multi-task-elastic-net" title="Permalink to this heading">¶</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.MultiTaskElasticNet.html#sklearn.linear_model.MultiTaskElasticNet" title="sklearn.linear_model.MultiTaskElasticNet"><code class="xref py py-class docutils literal notranslate"><span class="pre">MultiTaskElasticNet</span></code></a> is an elastic-net model that estimates sparse
coefficients for multiple regression problems jointly: <code class="docutils literal notranslate"><span class="pre">Y</span></code> is a 2D array
of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_tasks)</span></code>. The constraint is that the selected
features are the same for all the regression problems, also called tasks.</p>
<p>Mathematically, it consists of a linear model trained with a mixed
<span class="math notranslate nohighlight">\(\ell_1\)</span> <span class="math notranslate nohighlight">\(\ell_2\)</span>-norm and <span class="math notranslate nohighlight">\(\ell_2\)</span>-norm for regularization.
The objective function to minimize is:</p>
<div class="math notranslate nohighlight">
\[\min_{W} { \frac{1}{2n_{\text{samples}}} ||X W - Y||_{\text{Fro}}^2 + \alpha \rho ||W||_{2 1} +
\frac{\alpha(1-\rho)}{2} ||W||_{\text{Fro}}^2}\]</div>
<p>The implementation in the class <a class="reference internal" href="generated/sklearn.linear_model.MultiTaskElasticNet.html#sklearn.linear_model.MultiTaskElasticNet" title="sklearn.linear_model.MultiTaskElasticNet"><code class="xref py py-class docutils literal notranslate"><span class="pre">MultiTaskElasticNet</span></code></a> uses coordinate descent as
the algorithm to fit the coefficients.</p>
<p>The class <a class="reference internal" href="generated/sklearn.linear_model.MultiTaskElasticNetCV.html#sklearn.linear_model.MultiTaskElasticNetCV" title="sklearn.linear_model.MultiTaskElasticNetCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">MultiTaskElasticNetCV</span></code></a> can be used to set the parameters
<code class="docutils literal notranslate"><span class="pre">alpha</span></code> (<span class="math notranslate nohighlight">\(\alpha\)</span>) and <code class="docutils literal notranslate"><span class="pre">l1_ratio</span></code> (<span class="math notranslate nohighlight">\(\rho\)</span>) by cross-validation.</p>
</section>
<section id="least-angle-regression">
<span id="id11"></span><h2><span class="section-number">1.1.7. </span>Least Angle Regression<a class="headerlink" href="#least-angle-regression" title="Permalink to this heading">¶</a></h2>
<p>Least-angle regression (LARS) is a regression algorithm for
high-dimensional data, developed by Bradley Efron, Trevor Hastie, Iain
Johnstone and Robert Tibshirani. LARS is similar to forward stepwise
regression. At each step, it finds the feature most correlated with the
target. When there are multiple features having equal correlation, instead
of continuing along the same feature, it proceeds in a direction equiangular
between the features.</p>
<p>The advantages of LARS are:</p>
<blockquote>
<div><ul class="simple">
<li><p>It is numerically efficient in contexts where the number of features
is significantly greater than the number of samples.</p></li>
<li><p>It is computationally just as fast as forward selection and has
the same order of complexity as ordinary least squares.</p></li>
<li><p>It produces a full piecewise linear solution path, which is
useful in cross-validation or similar attempts to tune the model.</p></li>
<li><p>If two features are almost equally correlated with the target,
then their coefficients should increase at approximately the same
rate. The algorithm thus behaves as intuition would expect, and
also is more stable.</p></li>
<li><p>It is easily modified to produce solutions for other estimators,
like the Lasso.</p></li>
</ul>
</div></blockquote>
<p>The disadvantages of the LARS method include:</p>
<blockquote>
<div><ul class="simple">
<li><p>Because LARS is based upon an iterative refitting of the
residuals, it would appear to be especially sensitive to the
effects of noise. This problem is discussed in detail by Weisberg
in the discussion section of the Efron et al. (2004) Annals of
Statistics article.</p></li>
</ul>
</div></blockquote>
<p>The LARS model can be used via the estimator <a class="reference internal" href="generated/sklearn.linear_model.Lars.html#sklearn.linear_model.Lars" title="sklearn.linear_model.Lars"><code class="xref py py-class docutils literal notranslate"><span class="pre">Lars</span></code></a>, or its
low-level implementation <a class="reference internal" href="generated/sklearn.linear_model.lars_path.html#sklearn.linear_model.lars_path" title="sklearn.linear_model.lars_path"><code class="xref py py-func docutils literal notranslate"><span class="pre">lars_path</span></code></a> or <a class="reference internal" href="generated/sklearn.linear_model.lars_path_gram.html#sklearn.linear_model.lars_path_gram" title="sklearn.linear_model.lars_path_gram"><code class="xref py py-func docutils literal notranslate"><span class="pre">lars_path_gram</span></code></a>.</p>
</section>
<section id="lars-lasso">
<h2><span class="section-number">1.1.8. </span>LARS Lasso<a class="headerlink" href="#lars-lasso" title="Permalink to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.linear_model.LassoLars.html#sklearn.linear_model.LassoLars" title="sklearn.linear_model.LassoLars"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLars</span></code></a> is a lasso model implemented using the LARS
algorithm, and unlike the implementation based on coordinate descent,
this yields the exact solution, which is piecewise linear as a
function of the norm of its coefficients.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_lasso_lars.html"><img alt="../_images/sphx_glr_plot_lasso_lars_001.png" src="../_images/sphx_glr_plot_lasso_lars_001.png" style="width: 320.0px; height: 240.0px;" /></a>
</figure>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</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">LassoLars</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">.1</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">False</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">LassoLars(alpha=0.1, normalize=False)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.6..., 0. ])</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_lars.html#sphx-glr-auto-examples-linear-model-plot-lasso-lars-py"><span class="std std-ref">Lasso path using LARS</span></a></p></li>
</ul>
</aside>
<p>The Lars algorithm provides the full path of the coefficients along
the regularization parameter almost for free, thus a common operation
is to retrieve the path with one of the functions <a class="reference internal" href="generated/sklearn.linear_model.lars_path.html#sklearn.linear_model.lars_path" title="sklearn.linear_model.lars_path"><code class="xref py py-func docutils literal notranslate"><span class="pre">lars_path</span></code></a>
or <a class="reference internal" href="generated/sklearn.linear_model.lars_path_gram.html#sklearn.linear_model.lars_path_gram" title="sklearn.linear_model.lars_path_gram"><code class="xref py py-func docutils literal notranslate"><span class="pre">lars_path_gram</span></code></a>.</p>
<section id="mathematical-formulation">
<h3><span class="section-number">1.1.8.1. </span>Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Permalink to this heading">¶</a></h3>
<p>The algorithm is similar to forward stepwise regression, but instead
of including features at each step, the estimated coefficients are
increased in a direction equiangular to each one’s correlations with
the residual.</p>
<p>Instead of giving a vector result, the LARS solution consists of a
curve denoting the solution for each value of the <span class="math notranslate nohighlight">\(\ell_1\)</span> norm of the
parameter vector. The full coefficients path is stored in the array
<code class="docutils literal notranslate"><span class="pre">coef_path_</span></code> of shape <code class="docutils literal notranslate"><span class="pre">(n_features,</span> <span class="pre">max_features</span> <span class="pre">+</span> <span class="pre">1)</span></code>. The first
column is always zero.</p>
<aside class="topic">
<p class="topic-title">References:</p>
<ul class="simple">
<li><p>Original Algorithm is detailed in the paper <a class="reference external" href="https://www-stat.stanford.edu/~hastie/Papers/LARS/LeastAngle_2002.pdf">Least Angle Regression</a>
by Hastie et al.</p></li>
</ul>
</aside>
</section>
</section>
<section id="orthogonal-matching-pursuit-omp">
<span id="omp"></span><h2><span class="section-number">1.1.9. </span>Orthogonal Matching Pursuit (OMP)<a class="headerlink" href="#orthogonal-matching-pursuit-omp" title="Permalink to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.linear_model.OrthogonalMatchingPursuit.html#sklearn.linear_model.OrthogonalMatchingPursuit" title="sklearn.linear_model.OrthogonalMatchingPursuit"><code class="xref py py-class docutils literal notranslate"><span class="pre">OrthogonalMatchingPursuit</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.orthogonal_mp.html#sklearn.linear_model.orthogonal_mp" title="sklearn.linear_model.orthogonal_mp"><code class="xref py py-func docutils literal notranslate"><span class="pre">orthogonal_mp</span></code></a> implement the OMP
algorithm for approximating the fit of a linear model with constraints imposed
on the number of non-zero coefficients (ie. the <span class="math notranslate nohighlight">\(\ell_0\)</span> pseudo-norm).</p>
<p>Being a forward feature selection method like <a class="reference internal" href="#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a>,
orthogonal matching pursuit can approximate the optimum solution vector with a
fixed number of non-zero elements:</p>
<div class="math notranslate nohighlight">
\[\underset{w}{\operatorname{arg\,min\,}} ||y - Xw||_2^2 \text{ subject to } ||w||_0 \leq n_{\text{nonzero\_coefs}}\]</div>
<p>Alternatively, orthogonal matching pursuit can target a specific error instead
of a specific number of non-zero coefficients. This can be expressed as:</p>
<div class="math notranslate nohighlight">
\[\underset{w}{\operatorname{arg\,min\,}} ||w||_0 \text{ subject to } ||y-Xw||_2^2 \leq \text{tol}\]</div>
<p>OMP is based on a greedy algorithm that includes at each step the atom most
highly correlated with the current residual. It is similar to the simpler
matching pursuit (MP) method, but better in that at each iteration, the
residual is recomputed using an orthogonal projection on the space of the
previously chosen dictionary elements.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_omp.html#sphx-glr-auto-examples-linear-model-plot-omp-py"><span class="std std-ref">Orthogonal Matching Pursuit</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References:</p>
<ul class="simple">
<li><p><a class="reference external" href="https://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf">https://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf</a></p></li>
<li><p><a class="reference external" href="https://www.di.ens.fr/~mallat/papiers/MallatPursuit93.pdf">Matching pursuits with time-frequency dictionaries</a>,
S. G. Mallat, Z. Zhang,</p></li>
</ul>
</aside>
</section>
<section id="bayesian-regression">
<span id="id13"></span><h2><span class="section-number">1.1.10. </span>Bayesian Regression<a class="headerlink" href="#bayesian-regression" title="Permalink to this heading">¶</a></h2>
<p>Bayesian regression techniques can be used to include regularization
parameters in the estimation procedure: the regularization parameter is
not set in a hard sense but tuned to the data at hand.</p>
<p>This can be done by introducing <a class="reference external" href="https://en.wikipedia.org/wiki/Non-informative_prior#Uninformative_priors">uninformative priors</a>
over the hyper parameters of the model.
The <span class="math notranslate nohighlight">\(\ell_{2}\)</span> regularization used in <a class="reference internal" href="#ridge-regression"><span class="std std-ref">Ridge regression and classification</span></a> is
equivalent to finding a maximum a posteriori estimation under a Gaussian prior
over the coefficients <span class="math notranslate nohighlight">\(w\)</span> with precision <span class="math notranslate nohighlight">\(\lambda^{-1}\)</span>.
Instead of setting <code class="docutils literal notranslate"><span class="pre">lambda</span></code> manually, it is possible to treat it as a random
variable to be estimated from the data.</p>
<p>To obtain a fully probabilistic model, the output <span class="math notranslate nohighlight">\(y\)</span> is assumed
to be Gaussian distributed around <span class="math notranslate nohighlight">\(X w\)</span>:</p>
<div class="math notranslate nohighlight">
\[p(y|X,w,\alpha) = \mathcal{N}(y|X w,\alpha)\]</div>
<p>where <span class="math notranslate nohighlight">\(\alpha\)</span> is again treated as a random variable that is to be
estimated from the data.</p>
<p>The advantages of Bayesian Regression are:</p>
<blockquote>
<div><ul class="simple">
<li><p>It adapts to the data at hand.</p></li>
<li><p>It can be used to include regularization parameters in the
estimation procedure.</p></li>
</ul>
</div></blockquote>
<p>The disadvantages of Bayesian regression include:</p>
<blockquote>
<div><ul class="simple">
<li><p>Inference of the model can be time consuming.</p></li>
</ul>
</div></blockquote>
<aside class="topic">
<p class="topic-title">References</p>
<ul class="simple">
<li><p>A good introduction to Bayesian methods is given in C. Bishop: Pattern
Recognition and Machine learning</p></li>
<li><p>Original Algorithm is detailed in the book <code class="docutils literal notranslate"><span class="pre">Bayesian</span> <span class="pre">learning</span> <span class="pre">for</span> <span class="pre">neural</span>
<span class="pre">networks</span></code> by Radford M. Neal</p></li>
</ul>
</aside>
<section id="bayesian-ridge-regression">
<span id="id14"></span><h3><span class="section-number">1.1.10.1. </span>Bayesian Ridge Regression<a class="headerlink" href="#bayesian-ridge-regression" title="Permalink to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.linear_model.BayesianRidge.html#sklearn.linear_model.BayesianRidge" title="sklearn.linear_model.BayesianRidge"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianRidge</span></code></a> estimates a probabilistic model of the
regression problem as described above.
The prior for the coefficient <span class="math notranslate nohighlight">\(w\)</span> is given by a spherical Gaussian:</p>
<div class="math notranslate nohighlight">
\[p(w|\lambda) =
\mathcal{N}(w|0,\lambda^{-1}\mathbf{I}_{p})\]</div>
<p>The priors over <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\lambda\)</span> are chosen to be <a class="reference external" href="https://en.wikipedia.org/wiki/Gamma_distribution">gamma
distributions</a>, the
conjugate prior for the precision of the Gaussian. The resulting model is
called <em>Bayesian Ridge Regression</em>, and is similar to the classical
<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>.</p>
<p>The parameters <span class="math notranslate nohighlight">\(w\)</span>, <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\lambda\)</span> are estimated
jointly during the fit of the model, the regularization parameters
<span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\lambda\)</span> being estimated by maximizing the
<em>log marginal likelihood</em>. The scikit-learn implementation
is based on the algorithm described in Appendix A of (Tipping, 2001)
where the update of the parameters <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\lambda\)</span> is done
as suggested in (MacKay, 1992). The initial value of the maximization procedure
can be set with the hyperparameters <code class="docutils literal notranslate"><span class="pre">alpha_init</span></code> and <code class="docutils literal notranslate"><span class="pre">lambda_init</span></code>.</p>
<p>There are four more hyperparameters, <span class="math notranslate nohighlight">\(\alpha_1\)</span>, <span class="math notranslate nohighlight">\(\alpha_2\)</span>,
<span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span> of the gamma prior distributions over
<span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\lambda\)</span>. These are usually chosen to be
<em>non-informative</em>. By default <span class="math notranslate nohighlight">\(\alpha_1 = \alpha_2 = \lambda_1 = \lambda_2 = 10^{-6}\)</span>.</p>
<p>Bayesian Ridge Regression is used for regression:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">]</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">BayesianRidge</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="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="go">BayesianRidge()</span>
</pre></div>
</div>
<p>After being fitted, the model can then be used to predict new values:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></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="mf">0.</span><span class="p">]])</span>
<span class="go">array([0.50000013])</span>
</pre></div>
</div>
<p>The coefficients <span class="math notranslate nohighlight">\(w\)</span> of the model can be accessed:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.49999993, 0.49999993])</span>
</pre></div>
</div>
<p>Due to the Bayesian framework, the weights found are slightly different to the
ones found by <a class="reference internal" href="#ordinary-least-squares"><span class="std std-ref">Ordinary Least Squares</span></a>. However, Bayesian Ridge Regression
is more robust to ill-posed problems.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_bayesian_ridge_curvefit.html#sphx-glr-auto-examples-linear-model-plot-bayesian-ridge-curvefit-py"><span class="std std-ref">Curve Fitting with Bayesian Ridge Regression</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References:</p>
<ul class="simple">
<li><p>Section 3.3 in Christopher M. Bishop: Pattern Recognition and Machine Learning, 2006</p></li>
<li><p>David J. C. MacKay, <a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.9072&rep=rep1&type=pdf">Bayesian Interpolation</a>, 1992.</p></li>
<li><p>Michael E. Tipping, <a class="reference external" href="http://www.jmlr.org/papers/volume1/tipping01a/tipping01a.pdf">Sparse Bayesian Learning and the Relevance Vector Machine</a>, 2001.</p></li>
</ul>
</aside>
</section>
<section id="automatic-relevance-determination-ard">
<span id="automatic-relevance-determination"></span><h3><span class="section-number">1.1.10.2. </span>Automatic Relevance Determination - ARD<a class="headerlink" href="#automatic-relevance-determination-ard" title="Permalink to this heading">¶</a></h3>
<p>The Automatic Relevance Determination (as being implemented in
<a class="reference internal" href="generated/sklearn.linear_model.ARDRegression.html#sklearn.linear_model.ARDRegression" title="sklearn.linear_model.ARDRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">ARDRegression</span></code></a>) is a kind of linear model which is very similar to the
<a class="reference internal" href="#id14">Bayesian Ridge Regression</a>, but that leads to sparser coefficients <span class="math notranslate nohighlight">\(w\)</span>
<a class="footnote-reference brackets" href="#id19" id="id15" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id20" id="id16" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a>.</p>
<p><a class="reference internal" href="generated/sklearn.linear_model.ARDRegression.html#sklearn.linear_model.ARDRegression" title="sklearn.linear_model.ARDRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">ARDRegression</span></code></a> poses a different prior over <span class="math notranslate nohighlight">\(w\)</span>: it drops
the spherical Gaussian distribution for a centered elliptic Gaussian
distribution. This means each coefficient <span class="math notranslate nohighlight">\(w_{i}\)</span> can itself be drawn from
a Gaussian distribution, centered on zero and with a precision
<span class="math notranslate nohighlight">\(\lambda_{i}\)</span>:</p>
<div class="math notranslate nohighlight">
\[p(w|\lambda) = \mathcal{N}(w|0,A^{-1})\]</div>
<p>with <span class="math notranslate nohighlight">\(A\)</span> being a positive definite diagonal matrix and
<span class="math notranslate nohighlight">\(\text{diag}(A) = \lambda = \{\lambda_{1},...,\lambda_{p}\}\)</span>.</p>
<p>In contrast to the <a class="reference internal" href="#id14">Bayesian Ridge Regression</a>, each coordinate of
<span class="math notranslate nohighlight">\(w_{i}\)</span> has its own standard deviation <span class="math notranslate nohighlight">\(\frac{1}{\lambda_i}\)</span>. The
prior over all <span class="math notranslate nohighlight">\(\lambda_i\)</span> is chosen to be the same gamma distribution
given by the hyperparameters <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span>.</p>
<p>ARD is also known in the literature as <em>Sparse Bayesian Learning</em> and <em>Relevance
Vector Machine</em> <a class="footnote-reference brackets" href="#id21" id="id17" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a> <a class="footnote-reference brackets" href="#id23" id="id18" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>. For a worked-out comparison between ARD and <a class="reference internal" href="#id14">Bayesian
Ridge Regression</a>, see the example below.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ard.html#sphx-glr-auto-examples-linear-model-plot-ard-py"><span class="std std-ref">Comparing Linear Bayesian Regressors</span></a></p></li>
</ul>
</aside>
<aside class="topic">
<p class="topic-title">References:</p>
<aside class="footnote brackets" id="id19" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id15">1</a><span class="fn-bracket">]</span></span>
<p>Christopher M. Bishop: Pattern Recognition and Machine Learning, Chapter 7.2.1</p>
</aside>
<aside class="footnote brackets" id="id20" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id16">2</a><span class="fn-bracket">]</span></span>
<p>David Wipf and Srikantan Nagarajan: <a class="reference external" href="https://papers.nips.cc/paper/3372-a-new-view-of-automatic-relevance-determination.pdf">A new view of automatic relevance determination</a></p>
</aside>
<aside class="footnote brackets" id="id21" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id17">3</a><span class="fn-bracket">]</span></span>
<p>Michael E. Tipping: <a class="reference external" href="http://www.jmlr.org/papers/volume1/tipping01a/tipping01a.pdf">Sparse Bayesian Learning and the Relevance Vector Machine</a></p>
</aside>
<aside class="footnote brackets" id="id23" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id18">4</a><span class="fn-bracket">]</span></span>
<p>Tristan Fletcher: <a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.651.8603&rep=rep1&type=pdf">Relevance Vector Machines explained</a></p>
</aside>
</aside>
</section>
</section>
<section id="logistic-regression">
<span id="id24"></span><h2><span class="section-number">1.1.11. </span>Logistic regression<a class="headerlink" href="#logistic-regression" title="Permalink to this heading">¶</a></h2>
<p>Logistic regression, despite its name, is a linear model for classification
rather than regression. Logistic regression is also known in the literature as
logit regression, maximum-entropy classification (MaxEnt) or the log-linear
classifier. In this model, the probabilities describing the possible outcomes
of a single trial are modeled using a
<a class="reference external" href="https://en.wikipedia.org/wiki/Logistic_function">logistic function</a>.</p>
<p>Logistic regression is implemented in <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>.
This implementation can fit binary, One-vs-Rest, or multinomial logistic
regression with optional <span class="math notranslate nohighlight">\(\ell_1\)</span>, <span class="math notranslate nohighlight">\(\ell_2\)</span> or Elastic-Net
regularization.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Regularization is applied by default, which is common in machine
learning but not in statistics. Another advantage of regularization is
that it improves numerical stability. No regularization amounts to
setting C to a very high value.</p>
</div>
<section id="binary-case">
<h3><span class="section-number">1.1.11.1. </span>Binary Case<a class="headerlink" href="#binary-case" title="Permalink to this heading">¶</a></h3>
<p>For notational ease, we assume that the target <span class="math notranslate nohighlight">\(y_i\)</span> takes values in the
set <span class="math notranslate nohighlight">\(\{0, 1\}\)</span> for data point <span class="math notranslate nohighlight">\(i\)</span>.
Once fitted, the <a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression.predict_proba" title="sklearn.linear_model.LogisticRegression.predict_proba"><code class="xref py py-meth docutils literal notranslate"><span class="pre">predict_proba</span></code></a>
method of <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> predicts
the probability of the positive class <span class="math notranslate nohighlight">\(P(y_i=1|X_i)\)</span> as</p>
<div class="math notranslate nohighlight">
\[\hat{p}(X_i) = \operatorname{expit}(X_i w + w_0) = \frac{1}{1 + \exp(-X_i w - w_0)}.\]</div>
<p>As an optimization problem, binary
class logistic regression with regularization term <span class="math notranslate nohighlight">\(r(w)\)</span> minimizes the
following cost function:</p>
<div class="math notranslate nohighlight">
\[\min_{w} C \sum_{i=1}^n \left(-y_i \log(\hat{p}(X_i)) - (1 - y_i) \log(1 - \hat{p}(X_i))\right) + r(w).\]</div>
<p>We currently provide four choices for the regularization term <span class="math notranslate nohighlight">\(r(w)\)</span> via
the <code class="docutils literal notranslate"><span class="pre">penalty</span></code> argument:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p>penalty</p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(r(w)\)</span></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">None</span></code></p></td>
<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>