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<ul>
<li><a class="reference internal" href="#">1.1. Generalized 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="#ordinary-least-squares-complexity">1.1.1.1. Ordinary Least Squares Complexity</a></li>
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<li><a class="reference internal" href="#ridge-regression">1.1.2. Ridge Regression</a><ul>
<li><a class="reference internal" href="#ridge-complexity">1.1.2.1. Ridge Complexity</a></li>
<li><a class="reference internal" href="#setting-the-regularization-parameter-generalized-cross-validation">1.1.2.2. Setting the regularization parameter: generalized Cross-Validation</a></li>
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<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>
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<li><a class="reference internal" href="#automatic-relevance-determination-ard">1.1.10.2. Automatic Relevance Determination - ARD</a></li>
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<li><a class="reference internal" href="#logistic-regression">1.1.11. Logistic regression</a></li>
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<li><a class="reference internal" href="#polynomial-regression-extending-linear-models-with-basis-functions">1.1.16. Polynomial regression: extending linear models with basis functions</a></li>
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<div class="section" id="generalized-linear-models">
<span id="linear-model"></span><h1>1.1. Generalized Linear Models<a class="headerlink" href="#generalized-linear-models" title="Permalink to this headline">¶</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 input
variables. In mathematical notion, if <span class="math">\(\hat{y}\)</span> is the predicted
value.</p>
<div class="math">
\[\hat{y}(w, x) = w_0 + w_1 x_1 + ... + w_p x_p\]</div>
<p>Across the module, we designate the vector <span class="math">\(w = (w_1,
..., w_p)\)</span> as <code class="docutils literal"><span class="pre">coef_</span></code> and <span class="math">\(w_0\)</span> as <code class="docutils literal"><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>
<div class="section" id="ordinary-least-squares">
<span id="id1"></span><h2>1.1.1. Ordinary Least Squares<a class="headerlink" href="#ordinary-least-squares" title="Permalink to this headline">¶</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"><span class="pre">LinearRegression</span></code></a> fits a linear model with coefficients
<span class="math">\(w = (w_1, ..., w_p)\)</span> to minimize the residual sum
of squares between the observed responses in the dataset, and the
responses predicted by the linear approximation. Mathematically it
solves a problem of the form:</p>
<div class="math">
\[\underset{w}{min\,} {|| X w - y||_2}^2\]</div>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_ols.html"><img alt="../_images/sphx_glr_plot_ols_0011.png" src="../_images/sphx_glr_plot_ols_0011.png" style="width: 320.0px; height: 240.0px;" /></a>
</div>
<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"><span class="pre">LinearRegression</span></code></a> will take in its <code class="docutils literal"><span class="pre">fit</span></code> method arrays X, y
and will store the coefficients <span class="math">\(w\)</span> of the linear model in its
<code class="docutils literal"><span class="pre">coef_</span></code> member:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="k">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(copy_X=True, fit_intercept=True, n_jobs=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([0.5, 0.5])</span>
</pre></div>
</div>
<p>However, coefficient estimates for Ordinary Least Squares rely on the
independence of the model terms. When terms are correlated and the
columns of the design matrix <span class="math">\(X\)</span> have an approximate 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 response, producing a large
variance. This situation of <em>multicollinearity</em> can arise, for
example, when data are collected without an experimental design.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<div class="section" id="ordinary-least-squares-complexity">
<h3>1.1.1.1. Ordinary Least Squares Complexity<a class="headerlink" href="#ordinary-least-squares-complexity" title="Permalink to this headline">¶</a></h3>
<p>This method computes the least squares solution using a singular value
decomposition of X. If X is a matrix of size (n, p) this method has a
cost of <span class="math">\(O(n p^2)\)</span>, assuming that <span class="math">\(n \geq p\)</span>.</p>
</div>
</div>
<div class="section" id="ridge-regression">
<span id="id2"></span><h2>1.1.2. Ridge Regression<a class="headerlink" href="#ridge-regression" title="Permalink to this headline">¶</a></h2>
<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"><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
coefficients. The ridge coefficients minimize a penalized residual sum
of squares,</p>
<div class="math">
\[\underset{w}{min\,} {{|| X w - y||_2}^2 + \alpha {||w||_2}^2}\]</div>
<p>Here, <span class="math">\(\alpha \geq 0\)</span> is a complexity parameter that controls the amount
of shrinkage: the larger the value of <span class="math">\(\alpha\)</span>, the greater the amount
of shrinkage and thus the coefficients become more robust to collinearity.</p>
<div class="figure 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_0011.png" src="../_images/sphx_glr_plot_ridge_path_0011.png" style="width: 320.0px; height: 240.0px;" /></a>
</div>
<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"><span class="pre">Ridge</span></code></a> will take in its <code class="docutils literal"><span class="pre">fit</span></code> method
arrays X, y and will store the coefficients <span class="math">\(w\)</span> of the linear model in
its <code class="docutils literal"><span class="pre">coef_</span></code> member:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="k">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="o">.</span><span class="mi">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="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">Ridge(alpha=0.5, copy_X=True, fit_intercept=True, max_iter=None,</span>
<span class="go"> normalize=False, random_state=None, solver='auto', tol=0.001)</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>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><span class="xref std std-ref">sphx_glr_auto_examples_text_document_classification_20newsgroups.py</span></li>
</ul>
</div>
<div class="section" id="ridge-complexity">
<h3>1.1.2.1. Ridge Complexity<a class="headerlink" href="#ridge-complexity" title="Permalink to this headline">¶</a></h3>
<p>This method has the same order of complexity than an
<a class="reference internal" href="#ordinary-least-squares"><span class="std std-ref">Ordinary Least Squares</span></a>.</p>
</div>
<div class="section" id="setting-the-regularization-parameter-generalized-cross-validation">
<h3>1.1.2.2. Setting the regularization parameter: generalized Cross-Validation<a class="headerlink" href="#setting-the-regularization-parameter-generalized-cross-validation" title="Permalink to this headline">¶</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"><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 Generalized Cross-Validation
(GCV), an efficient form of leave-one-out cross-validation:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="k">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="p">[</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">10.0</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="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">RidgeCV(alphas=[0.1, 1.0, 10.0], cv=None, fit_intercept=True, scoring=None,</span>
<span class="go"> normalize=False)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">alpha_</span>
<span class="go">0.1</span>
</pre></div>
</div>
<div class="topic">
<p class="topic-title first">References</p>
<ul class="simple">
<li>“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="http://www.mit.edu/~9.520/spring07/Classes/rlsslides.pdf">course slides</a>).</li>
</ul>
</div>
</div>
</div>
<div class="section" id="lasso">
<span id="id3"></span><h2>1.1.3. Lasso<a class="headerlink" href="#lasso" title="Permalink to this headline">¶</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"><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 parameter values, effectively reducing the number of variables
upon which the given solution is dependent. For this reason, the Lasso
and its variants are fundamental to the field of compressed sensing.
Under certain conditions, it can recover the exact set of non-zero
weights (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 trained with <span class="math">\(\ell_1\)</span> prior
as regularizer. The objective function to minimize is:</p>
<div class="math">
\[\underset{w}{min\,} { \frac{1}{2n_{samples}} ||X w - y||_2 ^ 2 + \alpha ||w||_1}\]</div>
<p>The lasso estimate thus solves the minimization of the
least-squares penalty with <span class="math">\(\alpha ||w||_1\)</span> added, where
<span class="math">\(\alpha\)</span> is a constant and <span class="math">\(||w||_1\)</span> is the <span class="math">\(\ell_1\)</span>-norm of
the parameter 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"><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"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="k">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, copy_X=True, fit_intercept=True, max_iter=1000,</span>
<span class="go"> normalize=False, positive=False, precompute=False, random_state=None,</span>
<span class="go"> selection='cyclic', tol=0.0001, warm_start=False)</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>Also useful for lower-level tasks is 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"><span class="pre">lasso_path</span></code></a> that
computes the coefficients along the full path of possible values.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><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></li>
</ul>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p><strong>Feature selection with Lasso</strong></p>
<p class="last">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>
<div class="section" id="setting-regularization-parameter">
<h3>1.1.3.1. Setting regularization parameter<a class="headerlink" href="#setting-regularization-parameter" title="Permalink to this headline">¶</a></h3>
<p>The <code class="docutils literal"><span class="pre">alpha</span></code> parameter controls the degree of sparsity of the coefficients
estimated.</p>
<div class="section" id="using-cross-validation">
<h4>1.1.3.1.1. Using cross-validation<a class="headerlink" href="#using-cross-validation" title="Permalink to this headline">¶</a></h4>
<p>scikit-learn exposes objects that set the Lasso <code class="docutils literal"><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"><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"><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"><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 regressors,
<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"><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"><span class="pre">LassoLarsCV</span></code></a> has
the advantage of exploring more relevant values of <cite>alpha</cite> 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"><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_0021.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_0031.png" style="width: 307.2px; height: 230.39999999999998px;" /></a></strong></p></div>
<div class="section" id="information-criteria-based-model-selection">
<h4>1.1.3.1.2. Information-criteria based model selection<a class="headerlink" href="#information-criteria-based-model-selection" title="Permalink to this headline">¶</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"><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. However, such criteria needs a
proper estimation of the degrees of freedom of the solution, are
derived for large samples (asymptotic results) and assume the model
is correct, i.e. that the data are actually generated by this model.
They also tend to break when the problem is badly conditioned
(more features than samples).</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_lasso_model_selection.html"><img alt="../_images/sphx_glr_plot_lasso_model_selection_0011.png" src="../_images/sphx_glr_plot_lasso_model_selection_0011.png" style="width: 320.0px; height: 240.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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: Cross-Validation / AIC / BIC</span></a></li>
</ul>
</div>
</div>
<div class="section" id="comparison-with-the-regularization-parameter-of-svm">
<h4>1.1.3.1.3. Comparison with the regularization parameter of SVM<a class="headerlink" href="#comparison-with-the-regularization-parameter-of-svm" title="Permalink to this headline">¶</a></h4>
<p>The equivalence between <code class="docutils literal"><span class="pre">alpha</span></code> and the regularization parameter of SVM,
<code class="docutils literal"><span class="pre">C</span></code> is given by <code class="docutils literal"><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"><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>
</div>
</div>
</div>
<div class="section" id="multi-task-lasso">
<span id="id4"></span><h2>1.1.4. Multi-task Lasso<a class="headerlink" href="#multi-task-lasso" title="Permalink to this headline">¶</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"><span class="pre">MultiTaskLasso</span></code></a> is a linear model that estimates sparse
coefficients for multiple regression problems jointly: <code class="docutils literal"><span class="pre">y</span></code> is a 2D array,
of shape <code class="docutils literal"><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-zeros in W obtained
with a simple Lasso or a MultiTaskLasso. The Lasso estimates yields
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_0011.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_0021.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><div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<p>Mathematically, it consists of a linear model trained with a mixed
<span class="math">\(\ell_1\)</span> <span class="math">\(\ell_2\)</span> prior as regularizer.
The objective function to minimize is:</p>
<div class="math">
\[\underset{w}{min\,} { \frac{1}{2n_{samples}} ||X W - Y||_{Fro} ^ 2 + \alpha ||W||_{21}}\]</div>
<p>where <span class="math">\(Fro\)</span> indicates the Frobenius norm:</p>
<div class="math">
\[||A||_{Fro} = \sqrt{\sum_{ij} a_{ij}^2}\]</div>
<p>and <span class="math">\(\ell_1\)</span> <span class="math">\(\ell_2\)</span> reads:</p>
<div class="math">
\[||A||_{2 1} = \sum_i \sqrt{\sum_j a_{ij}^2}\]</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"><span class="pre">MultiTaskLasso</span></code></a> uses coordinate descent as
the algorithm to fit the coefficients.</p>
</div>
<div class="section" id="elastic-net">
<span id="id5"></span><h2>1.1.5. Elastic Net<a class="headerlink" href="#elastic-net" title="Permalink to this headline">¶</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"><span class="pre">ElasticNet</span></code></a> is a linear regression model trained with L1 and L2 prior
as regularizer. 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"><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"><span class="pre">Ridge</span></code></a>. We control the convex
combination of L1 and L2 using the <code class="docutils literal"><span class="pre">l1_ratio</span></code> parameter.</p>
<p>Elastic-net is useful when there are multiple features which 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 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">
\[\underset{w}{min\,} { \frac{1}{2n_{samples}} ||X w - y||_2 ^ 2 + \alpha \rho ||w||_1 +
\frac{\alpha(1-\rho)}{2} ||w||_2 ^ 2}\]</div>
<div class="figure 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_0011.png" src="../_images/sphx_glr_plot_lasso_coordinate_descent_path_0011.png" style="width: 320.0px; height: 240.0px;" /></a>
</div>
<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"><span class="pre">ElasticNetCV</span></code></a> can be used to set the parameters
<code class="docutils literal"><span class="pre">alpha</span></code> (<span class="math">\(\alpha\)</span>) and <code class="docutils literal"><span class="pre">l1_ratio</span></code> (<span class="math">\(\rho\)</span>) by cross-validation.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
<li><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></li>
</ul>
</div>
</div>
<div class="section" id="multi-task-elastic-net">
<span id="id6"></span><h2>1.1.6. Multi-task Elastic Net<a class="headerlink" href="#multi-task-elastic-net" title="Permalink to this headline">¶</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"><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"><span class="pre">Y</span></code> is a 2D array,
of shape <code class="docutils literal"><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">\(\ell_1\)</span> <span class="math">\(\ell_2\)</span> prior and <span class="math">\(\ell_2\)</span> prior as regularizer.
The objective function to minimize is:</p>
<div class="math">
\[\underset{W}{min\,} { \frac{1}{2n_{samples}} ||X W - Y||_{Fro}^2 + \alpha \rho ||W||_{2 1} +
\frac{\alpha(1-\rho)}{2} ||W||_{Fro}^2}\]</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"><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"><span class="pre">MultiTaskElasticNetCV</span></code></a> can be used to set the parameters
<code class="docutils literal"><span class="pre">alpha</span></code> (<span class="math">\(\alpha\)</span>) and <code class="docutils literal"><span class="pre">l1_ratio</span></code> (<span class="math">\(\rho\)</span>) by cross-validation.</p>
</div>
<div class="section" id="least-angle-regression">
<span id="id7"></span><h2>1.1.7. Least Angle Regression<a class="headerlink" href="#least-angle-regression" title="Permalink to this headline">¶</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 predictor most correlated with the
response. When there are multiple predictors having equal correlation, instead
of continuing along the same predictor, it proceeds in a direction equiangular
between the predictors.</p>
<p>The advantages of LARS are:</p>
<blockquote>
<div><ul class="simple">
<li>It is numerically efficient in contexts where p >> n (i.e., when the
number of dimensions is significantly greater than the number of
points)</li>
<li>It is computationally just as fast as forward selection and has
the same order of complexity as an ordinary least squares.</li>
<li>It produces a full piecewise linear solution path, which is
useful in cross-validation or similar attempts to tune the model.</li>
<li>If two variables are almost equally correlated with the response,
then their coefficients should increase at approximately the same
rate. The algorithm thus behaves as intuition would expect, and
also is more stable.</li>
<li>It is easily modified to produce solutions for other estimators,
like the Lasso.</li>
</ul>
</div></blockquote>
<p>The disadvantages of the LARS method include:</p>
<blockquote>
<div><ul class="simple">
<li>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.</li>
</ul>
</div></blockquote>
<p>The LARS model can be used using 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"><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"><span class="pre">lars_path</span></code></a>.</p>
</div>
<div class="section" id="lars-lasso">
<h2>1.1.8. LARS Lasso<a class="headerlink" href="#lars-lasso" title="Permalink to this headline">¶</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"><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>
<div class="figure 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_0011.png" src="../_images/sphx_glr_plot_lasso_lars_0011.png" style="width: 320.0px; height: 240.0px;" /></a>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="k">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="mi">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">LassoLars(alpha=0.1, copy_X=True, eps=..., fit_intercept=True,</span>
<span class="go"> fit_path=True, max_iter=500, normalize=True, positive=False,</span>
<span class="go"> precompute='auto', verbose=False)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.717157..., 0. ])</span>
</pre></div>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<p>The Lars algorithm provides the full path of the coefficients along
the regularization parameter almost for free, thus a common operation
consist of retrieving the path with function <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"><span class="pre">lars_path</span></code></a></p>
<div class="section" id="mathematical-formulation">
<h3>1.1.8.1. Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Permalink to this headline">¶</a></h3>
<p>The algorithm is similar to forward stepwise regression, but instead
of including variables at each step, the estimated parameters 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 L1 norm of the
parameter vector. The full coefficients path is stored in the array
<code class="docutils literal"><span class="pre">coef_path_</span></code>, which has size (n_features, max_features+1). The first
column is always zero.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li>Original Algorithm is detailed in the paper <a class="reference external" href="http://www-stat.stanford.edu/~hastie/Papers/LARS/LeastAngle_2002.pdf">Least Angle Regression</a>
by Hastie et al.</li>
</ul>
</div>
</div>
</div>
<div class="section" id="orthogonal-matching-pursuit-omp">
<span id="omp"></span><h2>1.1.9. Orthogonal Matching Pursuit (OMP)<a class="headerlink" href="#orthogonal-matching-pursuit-omp" title="Permalink to this headline">¶</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"><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"><span class="pre">orthogonal_mp</span></code></a> implements the OMP
algorithm for approximating the fit of a linear model with constraints imposed
on the number of non-zero coefficients (ie. the L <sub>0</sub> 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">
\[\text{arg\,min\,} ||y - X\gamma||_2^2 \text{ subject to } \
||\gamma||_0 \leq n_{nonzero\_coefs}\]</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">
\[\text{arg\,min\,} ||\gamma||_0 \text{ subject to } ||y-X\gamma||_2^2 \
\leq \text{tol}\]</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>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><a class="reference external" href="http://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf">http://www.cs.technion.ac.il/~ronrubin/Publications/KSVD-OMP-v2.pdf</a></li>
<li><a class="reference external" href="http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf">Matching pursuits with time-frequency dictionaries</a>,
S. G. Mallat, Z. Zhang,</li>
</ul>
</div>
</div>
<div class="section" id="bayesian-regression">
<span id="id9"></span><h2>1.1.10. Bayesian Regression<a class="headerlink" href="#bayesian-regression" title="Permalink to this headline">¶</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">\(\ell_{2}\)</span> regularization used in <a class="reference internal" href="#id2">Ridge Regression</a> is equivalent
to finding a maximum a posteriori estimation under a Gaussian prior over the
parameters <span class="math">\(w\)</span> with precision <span class="math">\(\lambda^{-1}\)</span>. Instead of setting
<cite>lambda</cite> 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">\(y\)</span> is assumed
to be Gaussian distributed around <span class="math">\(X w\)</span>:</p>
<div class="math">
\[p(y|X,w,\alpha) = \mathcal{N}(y|X w,\alpha)\]</div>
<p>Alpha 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>It adapts to the data at hand.</li>
<li>It can be used to include regularization parameters in the
estimation procedure.</li>
</ul>
</div></blockquote>
<p>The disadvantages of Bayesian regression include:</p>
<blockquote>
<div><ul class="simple">
<li>Inference of the model can be time consuming.</li>
</ul>
</div></blockquote>
<div class="topic">
<p class="topic-title first">References</p>
<ul class="simple">
<li>A good introduction to Bayesian methods is given in C. Bishop: Pattern
Recognition and Machine learning</li>
<li>Original Algorithm is detailed in the book <cite>Bayesian learning for neural
networks</cite> by Radford M. Neal</li>
</ul>
</div>
<div class="section" id="bayesian-ridge-regression">
<span id="id10"></span><h3>1.1.10.1. Bayesian Ridge Regression<a class="headerlink" href="#bayesian-ridge-regression" title="Permalink to this headline">¶</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"><span class="pre">BayesianRidge</span></code></a> estimates a probabilistic model of the
regression problem as described above.
The prior for the parameter <span class="math">\(w\)</span> is given by a spherical Gaussian:</p>
<div class="math">
\[p(w|\lambda) =
\mathcal{N}(w|0,\lambda^{-1}\bold{I_{p}})\]</div>
<p>The priors over <span class="math">\(\alpha\)</span> and <span class="math">\(\lambda\)</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.</p>
<p>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"><span class="pre">Ridge</span></code></a>. The parameters <span class="math">\(w\)</span>, <span class="math">\(\alpha\)</span> and
<span class="math">\(\lambda\)</span> are estimated jointly during the fit of the model. The
remaining hyperparameters are the parameters of the gamma priors over
<span class="math">\(\alpha\)</span> and <span class="math">\(\lambda\)</span>. These are usually chosen to be
<em>non-informative</em>. The parameters are estimated by maximizing the <em>marginal
log likelihood</em>.</p>
<p>By default <span class="math">\(\alpha_1 = \alpha_2 = \lambda_1 = \lambda_2 = 10^{-6}\)</span>.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_bayesian_ridge.html"><img alt="../_images/sphx_glr_plot_bayesian_ridge_0011.png" src="../_images/sphx_glr_plot_bayesian_ridge_0011.png" style="width: 300.0px; height: 250.0px;" /></a>
</div>
<p>Bayesian Ridge Regression is used for regression:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="k">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(alpha_1=1e-06, alpha_2=1e-06, compute_score=False, copy_X=True,</span>
<span class="go"> fit_intercept=True, lambda_1=1e-06, lambda_2=1e-06, n_iter=300,</span>
<span class="go"> normalize=False, tol=0.001, verbose=False)</span>
</pre></div>
</div>
<p>After being fitted, the model can then be used to predict new values:</p>
<div class="highlight-default"><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 weights <span class="math">\(w\)</span> of the model can be access:</p>
<div class="highlight-default"><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 problem.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/linear_model/plot_bayesian_ridge.html#sphx-glr-auto-examples-linear-model-plot-bayesian-ridge-py"><span class="std std-ref">Bayesian Ridge Regression</span></a></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References</p>
<ul class="simple">
<li>More details can be found in the article <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>
by MacKay, David J. C.</li>
</ul>
</div>
</div>
<div class="section" id="automatic-relevance-determination-ard">
<h3>1.1.10.2. Automatic Relevance Determination - ARD<a class="headerlink" href="#automatic-relevance-determination-ard" title="Permalink to this headline">¶</a></h3>
<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"><span class="pre">ARDRegression</span></code></a> is very similar to <a class="reference internal" href="#id10">Bayesian Ridge Regression</a>,
but can lead to sparser weights <span class="math">\(w\)</span> <a class="footnote-reference" href="#id15" id="id11">[1]</a> <a class="footnote-reference" href="#id16" id="id12">[2]</a>.
<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"><span class="pre">ARDRegression</span></code></a> poses a different prior over <span class="math">\(w\)</span>, by dropping the
assumption of the Gaussian being spherical.</p>
<p>Instead, the distribution over <span class="math">\(w\)</span> is assumed to be an axis-parallel,
elliptical Gaussian distribution.</p>
<p>This means each weight <span class="math">\(w_{i}\)</span> is drawn from a Gaussian distribution,
centered on zero and with a precision <span class="math">\(\lambda_{i}\)</span>:</p>
<div class="math">
\[p(w|\lambda) = \mathcal{N}(w|0,A^{-1})\]</div>
<p>with <span class="math">\(diag \; (A) = \lambda = \{\lambda_{1},...,\lambda_{p}\}\)</span>.</p>
<p>In contrast to <a class="reference internal" href="#id10">Bayesian Ridge Regression</a>, each coordinate of <span class="math">\(w_{i}\)</span>
has its own standard deviation <span class="math">\(\lambda_i\)</span>. The prior over all
<span class="math">\(\lambda_i\)</span> is chosen to be the same gamma distribution given by
hyperparameters <span class="math">\(\lambda_1\)</span> and <span class="math">\(\lambda_2\)</span>.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_ard.html"><img alt="../_images/sphx_glr_plot_ard_0011.png" src="../_images/sphx_glr_plot_ard_0011.png" style="width: 300.0px; height: 250.0px;" /></a>
</div>
<p>ARD is also known in the literature as <em>Sparse Bayesian Learning</em> and
<em>Relevance Vector Machine</em> <a class="footnote-reference" href="#id17" id="id13">[3]</a> <a class="footnote-reference" href="#id18" id="id14">[4]</a>.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><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">Automatic Relevance Determination Regression (ARD)</span></a></li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils footnote" frame="void" id="id15" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id11">[1]</a></td><td>Christopher M. Bishop: Pattern Recognition and Machine Learning, Chapter 7.2.1</td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id16" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id12">[2]</a></td><td>David Wipf and Srikantan Nagarajan: <a class="reference external" href="http://papers.nips.cc/paper/3372-a-new-view-of-automatic-relevance-determination.pdf">A new view of automatic relevance determination</a></td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id17" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id13">[3]</a></td><td>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></td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id18" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id14">[4]</a></td><td>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></td></tr>
</tbody>
</table>
</div>
</div>
</div>
<div class="section" id="logistic-regression">
<span id="id19"></span><h2>1.1.11. Logistic regression<a class="headerlink" href="#logistic-regression" title="Permalink to this headline">¶</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>The implementation of logistic regression in scikit-learn can be accessed from
class <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"><span class="pre">LogisticRegression</span></code></a>. This implementation can fit binary, One-vs-
Rest, or multinomial logistic regression with optional L2 or L1
regularization.</p>
<p>As an optimization problem, binary class L2 penalized logistic regression
minimizes the following cost function:</p>
<div class="math">
\[\underset{w, c}{min\,} \frac{1}{2}w^T w + C \sum_{i=1}^n \log(\exp(- y_i (X_i^T w + c)) + 1) .\]</div>
<p>Similarly, L1 regularized logistic regression solves the following
optimization problem</p>
<div class="math">
\[\underset{w, c}{min\,} \|w\|_1 + C \sum_{i=1}^n \log(\exp(- y_i (X_i^T w + c)) + 1).\]</div>
<p>Note that, in this notation, it’s assumed that the observation <span class="math">\(y_i\)</span> takes values in the set
<span class="math">\({-1, 1}\)</span> at trial <span class="math">\(i\)</span>.</p>
<p>The solvers implemented in the class <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"><span class="pre">LogisticRegression</span></code></a>
are “liblinear”, “newton-cg”, “lbfgs”, “sag” and “saga”:</p>
<p>The solver “liblinear” uses a coordinate descent (CD) algorithm, and relies
on the excellent C++ <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/liblinear/">LIBLINEAR library</a>, which is shipped with
scikit-learn. However, the CD algorithm implemented in liblinear cannot learn
a true multinomial (multiclass) model; instead, the optimization problem is
decomposed in a “one-vs-rest” fashion so separate binary classifiers are
trained for all classes. This happens under the hood, so
<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"><span class="pre">LogisticRegression</span></code></a> instances using this solver behave as multiclass
classifiers. For L1 penalization <a class="reference internal" href="generated/sklearn.svm.l1_min_c.html#sklearn.svm.l1_min_c" title="sklearn.svm.l1_min_c"><code class="xref py py-func docutils literal"><span class="pre">sklearn.svm.l1_min_c</span></code></a> allows to
calculate the lower bound for C in order to get a non “null” (all feature
weights to zero) model.</p>
<p>The “lbfgs”, “sag” and “newton-cg” solvers only support L2 penalization and
are found to converge faster for some high dimensional data. Setting
<cite>multi_class</cite> to “multinomial” with these solvers learns a true multinomial
logistic regression model <a class="footnote-reference" href="#id23" id="id20">[5]</a>, which means that its probability estimates
should be better calibrated than the default “one-vs-rest” setting.</p>
<p>The “sag” solver uses a Stochastic Average Gradient descent <a class="footnote-reference" href="#id24" id="id21">[6]</a>. It is faster
than other solvers for large datasets, when both the number of samples and the
number of features are large.</p>
<p>The “saga” solver <a class="footnote-reference" href="#id25" id="id22">[7]</a> is a variant of “sag” that also supports the
non-smooth <cite>penalty=”l1”</cite> option. This is therefore the solver of choice
for sparse multinomial logistic regression.</p>
<p>In a nutshell, one may choose the solver with the following rules:</p>
<table border="1" class="docutils">
<colgroup>
<col width="47%" />
<col width="53%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Case</th>
<th class="head">Solver</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td>L1 penalty</td>
<td>“liblinear” or “saga”</td>
</tr>
<tr class="row-odd"><td>Multinomial loss</td>
<td>“lbfgs”, “sag”, “saga” or “newton-cg”</td>
</tr>
<tr class="row-even"><td>Very Large dataset (<cite>n_samples</cite>)</td>
<td>“sag” or “saga”</td>
</tr>
</tbody>
</table>
<p>The “saga” solver is often the best choice. The “liblinear” solver is
used by default for historical reasons.</p>
<p>For large dataset, you may also consider using <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal"><span class="pre">SGDClassifier</span></code></a>
with ‘log’ loss.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/linear_model/plot_logistic_l1_l2_sparsity.html#sphx-glr-auto-examples-linear-model-plot-logistic-l1-l2-sparsity-py"><span class="std std-ref">L1 Penalty and Sparsity in Logistic Regression</span></a></li>
<li><a class="reference internal" href="../auto_examples/linear_model/plot_logistic_path.html#sphx-glr-auto-examples-linear-model-plot-logistic-path-py"><span class="std std-ref">Path with L1- Logistic Regression</span></a></li>
<li><a class="reference internal" href="../auto_examples/linear_model/plot_logistic_multinomial.html#sphx-glr-auto-examples-linear-model-plot-logistic-multinomial-py"><span class="std std-ref">Plot multinomial and One-vs-Rest Logistic Regression</span></a></li>
<li><a class="reference internal" href="../auto_examples/linear_model/plot_sparse_logistic_regression_20newsgroups.html#sphx-glr-auto-examples-linear-model-plot-sparse-logistic-regression-20newsgroups-py"><span class="std std-ref">Multiclass sparse logisitic regression on newgroups20</span></a></li>
<li><a class="reference internal" href="../auto_examples/linear_model/plot_sparse_logistic_regression_mnist.html#sphx-glr-auto-examples-linear-model-plot-sparse-logistic-regression-mnist-py"><span class="std std-ref">MNIST classfification using multinomial logistic + L1</span></a></li>
</ul>
</div>
<div class="topic" id="liblinear-differences">
<p class="topic-title first">Differences from liblinear:</p>
<p>There might be a difference in the scores obtained between
<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"><span class="pre">LogisticRegression</span></code></a> with <code class="docutils literal"><span class="pre">solver=liblinear</span></code>
or <code class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></code> and the external liblinear library directly,
when <code class="docutils literal"><span class="pre">fit_intercept=False</span></code> and the fit <code class="docutils literal"><span class="pre">coef_</span></code> (or) the data to
be predicted are zeroes. This is because for the sample(s) with
<code class="docutils literal"><span class="pre">decision_function</span></code> zero, <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"><span class="pre">LogisticRegression</span></code></a> and <code class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></code>
predict the negative class, while liblinear predicts the positive class.
Note that a model with <code class="docutils literal"><span class="pre">fit_intercept=False</span></code> and having many samples with
<code class="docutils literal"><span class="pre">decision_function</span></code> zero, is likely to be a underfit, bad model and you are
advised to set <code class="docutils literal"><span class="pre">fit_intercept=True</span></code> and increase the intercept_scaling.</p>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p><strong>Feature selection with sparse logistic regression</strong></p>
<p class="last">A logistic regression with L1 penalty yields sparse models, and 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><a class="reference internal" href="generated/sklearn.linear_model.LogisticRegressionCV.html#sklearn.linear_model.LogisticRegressionCV" title="sklearn.linear_model.LogisticRegressionCV"><code class="xref py py-class docutils literal"><span class="pre">LogisticRegressionCV</span></code></a> implements Logistic Regression with
builtin cross-validation to find out the optimal C parameter.
“newton-cg”, “sag”, “saga” and “lbfgs” solvers are found to be faster
for high-dimensional dense data, due to warm-starting. For the
multiclass case, if <cite>multi_class</cite> option is set to “ovr”, an optimal C
is obtained for each class and if the <cite>multi_class</cite> option is set to
“multinomial”, an optimal C is obtained by minimizing the cross-entropy
loss.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils footnote" frame="void" id="id23" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id20">[5]</a></td><td>Christopher M. Bishop: Pattern Recognition and Machine Learning, Chapter 4.3.4</td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id24" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id21">[6]</a></td><td>Mark Schmidt, Nicolas Le Roux, and Francis Bach: <a class="reference external" href="https://hal.inria.fr/hal-00860051/document">Minimizing Finite Sums with the Stochastic Average Gradient.</a></td></tr>
</tbody>
</table>
<table class="docutils footnote" frame="void" id="id25" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id22">[7]</a></td><td>Aaron Defazio, Francis Bach, Simon Lacoste-Julien: <a class="reference external" href="https://arxiv.org/abs/1407.0202">SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives.</a></td></tr>
</tbody>
</table>
</div>
</div>
<div class="section" id="stochastic-gradient-descent-sgd">
<h2>1.1.12. Stochastic Gradient Descent - SGD<a class="headerlink" href="#stochastic-gradient-descent-sgd" title="Permalink to this headline">¶</a></h2>
<p>Stochastic gradient descent is a simple yet very efficient approach
to fit linear models. It is particularly useful when the number of samples
(and the number of features) is very large.
The <code class="docutils literal"><span class="pre">partial_fit</span></code> method allows online/out-of-core learning.</p>
<p>The classes <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal"><span class="pre">SGDClassifier</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.SGDRegressor.html#sklearn.linear_model.SGDRegressor" title="sklearn.linear_model.SGDRegressor"><code class="xref py py-class docutils literal"><span class="pre">SGDRegressor</span></code></a> provide
functionality to fit linear models for classification and regression
using different (convex) loss functions and different penalties.
E.g., with <code class="docutils literal"><span class="pre">loss="log"</span></code>, <a class="reference internal" href="generated/sklearn.linear_model.SGDClassifier.html#sklearn.linear_model.SGDClassifier" title="sklearn.linear_model.SGDClassifier"><code class="xref py py-class docutils literal"><span class="pre">SGDClassifier</span></code></a>
fits a logistic regression model,
while with <code class="docutils literal"><span class="pre">loss="hinge"</span></code> it fits a linear support vector machine (SVM).</p>
<div class="topic">
<p class="topic-title first">References</p>
<ul class="simple">
<li><a class="reference internal" href="sgd.html#sgd"><span class="std std-ref">Stochastic Gradient Descent</span></a></li>
</ul>
</div>
</div>
<div class="section" id="perceptron">
<span id="id26"></span><h2>1.1.13. Perceptron<a class="headerlink" href="#perceptron" title="Permalink to this headline">¶</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.Perceptron.html#sklearn.linear_model.Perceptron" title="sklearn.linear_model.Perceptron"><code class="xref py py-class docutils literal"><span class="pre">Perceptron</span></code></a> is another simple classification algorithm suitable for
large scale learning. By default:</p>
<blockquote>
<div><ul class="simple">
<li>It does not require a learning rate.</li>
<li>It is not regularized (penalized).</li>
<li>It updates its model only on mistakes.</li>
</ul>
</div></blockquote>
<p>The last characteristic implies that the Perceptron is slightly faster to
train than SGD with the hinge loss and that the resulting models are
sparser.</p>
</div>
<div class="section" id="passive-aggressive-algorithms">
<span id="passive-aggressive"></span><h2>1.1.14. Passive Aggressive Algorithms<a class="headerlink" href="#passive-aggressive-algorithms" title="Permalink to this headline">¶</a></h2>
<p>The passive-aggressive algorithms are a family of algorithms for large-scale
learning. They are similar to the Perceptron in that they do not require a
learning rate. However, contrary to the Perceptron, they include a
regularization parameter <code class="docutils literal"><span class="pre">C</span></code>.</p>
<p>For classification, <a class="reference internal" href="generated/sklearn.linear_model.PassiveAggressiveClassifier.html#sklearn.linear_model.PassiveAggressiveClassifier" title="sklearn.linear_model.PassiveAggressiveClassifier"><code class="xref py py-class docutils literal"><span class="pre">PassiveAggressiveClassifier</span></code></a> can be used with
<code class="docutils literal"><span class="pre">loss='hinge'</span></code> (PA-I) or <code class="docutils literal"><span class="pre">loss='squared_hinge'</span></code> (PA-II). For regression,