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<ul>
<li><a class="reference internal" href="#">1.13. Feature selection</a><ul>
<li><a class="reference internal" href="#removing-features-with-low-variance">1.13.1. Removing features with low variance</a></li>
<li><a class="reference internal" href="#univariate-feature-selection">1.13.2. Univariate feature selection</a></li>
<li><a class="reference internal" href="#recursive-feature-elimination">1.13.3. Recursive feature elimination</a></li>
<li><a class="reference internal" href="#feature-selection-using-selectfrommodel">1.13.4. Feature selection using SelectFromModel</a><ul>
<li><a class="reference internal" href="#l1-based-feature-selection">1.13.4.1. L1-based feature selection</a></li>
<li><a class="reference internal" href="#tree-based-feature-selection">1.13.4.2. Tree-based feature selection</a></li>
</ul>
</li>
<li><a class="reference internal" href="#sequential-feature-selection">1.13.5. Sequential Feature Selection</a></li>
<li><a class="reference internal" href="#feature-selection-as-part-of-a-pipeline">1.13.6. Feature selection as part of a pipeline</a></li>
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<section id="feature-selection">
<span id="id1"></span><h1><span class="section-number">1.13. </span>Feature selection<a class="headerlink" href="#feature-selection" title="Link to this heading">¶</a></h1>
<p>The classes in the <a class="reference internal" href="classes.html#module-sklearn.feature_selection" title="sklearn.feature_selection"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.feature_selection</span></code></a> module can be used
for feature selection/dimensionality reduction on sample sets, either to
improve estimators’ accuracy scores or to boost their performance on very
high-dimensional datasets.</p>
<section id="removing-features-with-low-variance">
<span id="variance-threshold"></span><h2><span class="section-number">1.13.1. </span>Removing features with low variance<a class="headerlink" href="#removing-features-with-low-variance" title="Link to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.feature_selection.VarianceThreshold.html#sklearn.feature_selection.VarianceThreshold" title="sklearn.feature_selection.VarianceThreshold"><code class="xref py py-class docutils literal notranslate"><span class="pre">VarianceThreshold</span></code></a> is a simple baseline approach to feature selection.
It removes all features whose variance doesn’t meet some threshold.
By default, it removes all zero-variance features,
i.e. features that have the same value in all samples.</p>
<p>As an example, suppose that we have a dataset with boolean features,
and we want to remove all features that are either one or zero (on or off)
in more than 80% of the samples.
Boolean features are Bernoulli random variables,
and the variance of such variables is given by</p>
<div class="math notranslate nohighlight">
\[\mathrm{Var}[X] = p(1 - p)\]</div>
<p>so we can select using the threshold <code class="docutils literal notranslate"><span class="pre">.8</span> <span class="pre">*</span> <span class="pre">(1</span> <span class="pre">-</span> <span class="pre">.8)</span></code>:</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.feature_selection</span> <span class="kn">import</span> <span class="n">VarianceThreshold</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</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="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="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</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">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="mi">0</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">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">sel</span> <span class="o">=</span> <span class="n">VarianceThreshold</span><span class="p">(</span><span class="n">threshold</span><span class="o">=</span><span class="p">(</span><span class="mf">.8</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mf">.8</span><span class="p">)))</span>
<span class="gp">>>> </span><span class="n">sel</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([[0, 1],</span>
<span class="go"> [1, 0],</span>
<span class="go"> [0, 0],</span>
<span class="go"> [1, 1],</span>
<span class="go"> [1, 0],</span>
<span class="go"> [1, 1]])</span>
</pre></div>
</div>
<p>As expected, <code class="docutils literal notranslate"><span class="pre">VarianceThreshold</span></code> has removed the first column,
which has a probability <span class="math notranslate nohighlight">\(p = 5/6 > .8\)</span> of containing a zero.</p>
</section>
<section id="univariate-feature-selection">
<span id="id2"></span><h2><span class="section-number">1.13.2. </span>Univariate feature selection<a class="headerlink" href="#univariate-feature-selection" title="Link to this heading">¶</a></h2>
<p>Univariate feature selection works by selecting the best features based on
univariate statistical tests. It can be seen as a preprocessing step
to an estimator. Scikit-learn exposes feature selection routines
as objects that implement the <code class="docutils literal notranslate"><span class="pre">transform</span></code> method:</p>
<blockquote>
<div><ul class="simple">
<li><p><a class="reference internal" href="generated/sklearn.feature_selection.SelectKBest.html#sklearn.feature_selection.SelectKBest" title="sklearn.feature_selection.SelectKBest"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectKBest</span></code></a> removes all but the <span class="math notranslate nohighlight">\(k\)</span> highest scoring features</p></li>
<li><p><a class="reference internal" href="generated/sklearn.feature_selection.SelectPercentile.html#sklearn.feature_selection.SelectPercentile" title="sklearn.feature_selection.SelectPercentile"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectPercentile</span></code></a> removes all but a user-specified highest scoring
percentage of features</p></li>
<li><p>using common univariate statistical tests for each feature:
false positive rate <a class="reference internal" href="generated/sklearn.feature_selection.SelectFpr.html#sklearn.feature_selection.SelectFpr" title="sklearn.feature_selection.SelectFpr"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFpr</span></code></a>, false discovery rate
<a class="reference internal" href="generated/sklearn.feature_selection.SelectFdr.html#sklearn.feature_selection.SelectFdr" title="sklearn.feature_selection.SelectFdr"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFdr</span></code></a>, or family wise error <a class="reference internal" href="generated/sklearn.feature_selection.SelectFwe.html#sklearn.feature_selection.SelectFwe" title="sklearn.feature_selection.SelectFwe"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFwe</span></code></a>.</p></li>
<li><p><a class="reference internal" href="generated/sklearn.feature_selection.GenericUnivariateSelect.html#sklearn.feature_selection.GenericUnivariateSelect" title="sklearn.feature_selection.GenericUnivariateSelect"><code class="xref py py-class docutils literal notranslate"><span class="pre">GenericUnivariateSelect</span></code></a> allows to perform univariate feature
selection with a configurable strategy. This allows to select the best
univariate selection strategy with hyper-parameter search estimator.</p></li>
</ul>
</div></blockquote>
<p>For instance, we can use a F-test to retrieve the two
best features for a dataset as follows:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.feature_selection</span> <span class="kn">import</span> <span class="n">SelectKBest</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.feature_selection</span> <span class="kn">import</span> <span class="n">f_classif</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(150, 4)</span>
<span class="gp">>>> </span><span class="n">X_new</span> <span class="o">=</span> <span class="n">SelectKBest</span><span class="p">(</span><span class="n">f_classif</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">fit_transform</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="gp">>>> </span><span class="n">X_new</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(150, 2)</span>
</pre></div>
</div>
<p>These objects take as input a scoring function that returns univariate scores
and p-values (or only scores for <a class="reference internal" href="generated/sklearn.feature_selection.SelectKBest.html#sklearn.feature_selection.SelectKBest" title="sklearn.feature_selection.SelectKBest"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectKBest</span></code></a> and
<a class="reference internal" href="generated/sklearn.feature_selection.SelectPercentile.html#sklearn.feature_selection.SelectPercentile" title="sklearn.feature_selection.SelectPercentile"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectPercentile</span></code></a>):</p>
<blockquote>
<div><ul class="simple">
<li><p>For regression: <a class="reference internal" href="generated/sklearn.feature_selection.r_regression.html#sklearn.feature_selection.r_regression" title="sklearn.feature_selection.r_regression"><code class="xref py py-func docutils literal notranslate"><span class="pre">r_regression</span></code></a>, <a class="reference internal" href="generated/sklearn.feature_selection.f_regression.html#sklearn.feature_selection.f_regression" title="sklearn.feature_selection.f_regression"><code class="xref py py-func docutils literal notranslate"><span class="pre">f_regression</span></code></a>, <a class="reference internal" href="generated/sklearn.feature_selection.mutual_info_regression.html#sklearn.feature_selection.mutual_info_regression" title="sklearn.feature_selection.mutual_info_regression"><code class="xref py py-func docutils literal notranslate"><span class="pre">mutual_info_regression</span></code></a></p></li>
<li><p>For classification: <a class="reference internal" href="generated/sklearn.feature_selection.chi2.html#sklearn.feature_selection.chi2" title="sklearn.feature_selection.chi2"><code class="xref py py-func docutils literal notranslate"><span class="pre">chi2</span></code></a>, <a class="reference internal" href="generated/sklearn.feature_selection.f_classif.html#sklearn.feature_selection.f_classif" title="sklearn.feature_selection.f_classif"><code class="xref py py-func docutils literal notranslate"><span class="pre">f_classif</span></code></a>, <a class="reference internal" href="generated/sklearn.feature_selection.mutual_info_classif.html#sklearn.feature_selection.mutual_info_classif" title="sklearn.feature_selection.mutual_info_classif"><code class="xref py py-func docutils literal notranslate"><span class="pre">mutual_info_classif</span></code></a></p></li>
</ul>
</div></blockquote>
<p>The methods based on F-test estimate the degree of linear dependency between
two random variables. On the other hand, mutual information methods can capture
any kind of statistical dependency, but being nonparametric, they require more
samples for accurate estimation. Note that the <span class="math notranslate nohighlight">\(\chi^2\)</span>-test should only be
applied to non-negative features, such as frequencies.</p>
<aside class="topic">
<p class="topic-title">Feature selection with sparse data</p>
<p>If you use sparse data (i.e. data represented as sparse matrices),
<a class="reference internal" href="generated/sklearn.feature_selection.chi2.html#sklearn.feature_selection.chi2" title="sklearn.feature_selection.chi2"><code class="xref py py-func docutils literal notranslate"><span class="pre">chi2</span></code></a>, <a class="reference internal" href="generated/sklearn.feature_selection.mutual_info_regression.html#sklearn.feature_selection.mutual_info_regression" title="sklearn.feature_selection.mutual_info_regression"><code class="xref py py-func docutils literal notranslate"><span class="pre">mutual_info_regression</span></code></a>, <a class="reference internal" href="generated/sklearn.feature_selection.mutual_info_classif.html#sklearn.feature_selection.mutual_info_classif" title="sklearn.feature_selection.mutual_info_classif"><code class="xref py py-func docutils literal notranslate"><span class="pre">mutual_info_classif</span></code></a>
will deal with the data without making it dense.</p>
</aside>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Beware not to use a regression scoring function with a classification
problem, you will get useless results.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The <a class="reference internal" href="generated/sklearn.feature_selection.SelectPercentile.html#sklearn.feature_selection.SelectPercentile" title="sklearn.feature_selection.SelectPercentile"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectPercentile</span></code></a> and <a class="reference internal" href="generated/sklearn.feature_selection.SelectKBest.html#sklearn.feature_selection.SelectKBest" title="sklearn.feature_selection.SelectKBest"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectKBest</span></code></a> support unsupervised
feature selection as well. One needs to provide a <code class="docutils literal notranslate"><span class="pre">score_func</span></code> where <code class="docutils literal notranslate"><span class="pre">y=None</span></code>.
The <code class="docutils literal notranslate"><span class="pre">score_func</span></code> should use internally <code class="docutils literal notranslate"><span class="pre">X</span></code> to compute the scores.</p>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/feature_selection/plot_feature_selection.html#sphx-glr-auto-examples-feature-selection-plot-feature-selection-py"><span class="std std-ref">Univariate Feature Selection</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/feature_selection/plot_f_test_vs_mi.html#sphx-glr-auto-examples-feature-selection-plot-f-test-vs-mi-py"><span class="std std-ref">Comparison of F-test and mutual information</span></a></p></li>
</ul>
</aside>
</section>
<section id="recursive-feature-elimination">
<span id="rfe"></span><h2><span class="section-number">1.13.3. </span>Recursive feature elimination<a class="headerlink" href="#recursive-feature-elimination" title="Link to this heading">¶</a></h2>
<p>Given an external estimator that assigns weights to features (e.g., the
coefficients of a linear model), the goal of recursive feature elimination (<a class="reference internal" href="generated/sklearn.feature_selection.RFE.html#sklearn.feature_selection.RFE" title="sklearn.feature_selection.RFE"><code class="xref py py-class docutils literal notranslate"><span class="pre">RFE</span></code></a>)
is to select features by recursively considering smaller and smaller sets of
features. First, the estimator is trained on the initial set of features and
the importance of each feature is obtained either through any specific attribute
(such as <code class="docutils literal notranslate"><span class="pre">coef_</span></code>, <code class="docutils literal notranslate"><span class="pre">feature_importances_</span></code>) or callable. Then, the least important
features are pruned from current set of features. That procedure is recursively
repeated on the pruned set until the desired number of features to select is
eventually reached.</p>
<p><a class="reference internal" href="generated/sklearn.feature_selection.RFECV.html#sklearn.feature_selection.RFECV" title="sklearn.feature_selection.RFECV"><code class="xref py py-class docutils literal notranslate"><span class="pre">RFECV</span></code></a> performs RFE in a cross-validation loop to find the optimal
number of features. In more details, the number of features selected is tuned
automatically by fitting an <a class="reference internal" href="generated/sklearn.feature_selection.RFE.html#sklearn.feature_selection.RFE" title="sklearn.feature_selection.RFE"><code class="xref py py-class docutils literal notranslate"><span class="pre">RFE</span></code></a> selector on the different
cross-validation splits (provided by the <code class="docutils literal notranslate"><span class="pre">cv</span></code> parameter). The performance
of the <a class="reference internal" href="generated/sklearn.feature_selection.RFE.html#sklearn.feature_selection.RFE" title="sklearn.feature_selection.RFE"><code class="xref py py-class docutils literal notranslate"><span class="pre">RFE</span></code></a> selector are evaluated using <code class="docutils literal notranslate"><span class="pre">scorer</span></code> for different number
of selected features and aggregated together. Finally, the scores are averaged
across folds and the number of features selected is set to the number of
features that maximize the cross-validation score.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/feature_selection/plot_rfe_digits.html#sphx-glr-auto-examples-feature-selection-plot-rfe-digits-py"><span class="std std-ref">Recursive feature elimination</span></a>: A recursive feature elimination example
showing the relevance of pixels in a digit classification task.</p></li>
<li><p><a class="reference internal" href="../auto_examples/feature_selection/plot_rfe_with_cross_validation.html#sphx-glr-auto-examples-feature-selection-plot-rfe-with-cross-validation-py"><span class="std std-ref">Recursive feature elimination with cross-validation</span></a>: A recursive feature
elimination example with automatic tuning of the number of features
selected with cross-validation.</p></li>
</ul>
</aside>
</section>
<section id="feature-selection-using-selectfrommodel">
<span id="select-from-model"></span><h2><span class="section-number">1.13.4. </span>Feature selection using SelectFromModel<a class="headerlink" href="#feature-selection-using-selectfrommodel" title="Link to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.feature_selection.SelectFromModel.html#sklearn.feature_selection.SelectFromModel" title="sklearn.feature_selection.SelectFromModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFromModel</span></code></a> is a meta-transformer that can be used alongside any
estimator that assigns importance to each feature through a specific attribute (such as
<code class="docutils literal notranslate"><span class="pre">coef_</span></code>, <code class="docutils literal notranslate"><span class="pre">feature_importances_</span></code>) or via an <code class="docutils literal notranslate"><span class="pre">importance_getter</span></code> callable after fitting.
The features are considered unimportant and removed if the corresponding
importance of the feature values are below the provided
<code class="docutils literal notranslate"><span class="pre">threshold</span></code> parameter. Apart from specifying the threshold numerically,
there are built-in heuristics for finding a threshold using a string argument.
Available heuristics are “mean”, “median” and float multiples of these like
“0.1*mean”. In combination with the <code class="docutils literal notranslate"><span class="pre">threshold</span></code> criteria, one can use the
<code class="docutils literal notranslate"><span class="pre">max_features</span></code> parameter to set a limit on the number of features to select.</p>
<p>For examples on how it is to be used refer to the sections below.</p>
<aside class="topic">
<p class="topic-title">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/feature_selection/plot_select_from_model_diabetes.html#sphx-glr-auto-examples-feature-selection-plot-select-from-model-diabetes-py"><span class="std std-ref">Model-based and sequential feature selection</span></a></p></li>
</ul>
</aside>
<section id="l1-based-feature-selection">
<span id="l1-feature-selection"></span><h3><span class="section-number">1.13.4.1. </span>L1-based feature selection<a class="headerlink" href="#l1-based-feature-selection" title="Link to this heading">¶</a></h3>
<p><a class="reference internal" href="linear_model.html#linear-model"><span class="std std-ref">Linear models</span></a> penalized with the L1 norm have
sparse solutions: many of their estimated coefficients are zero. When the goal
is to reduce the dimensionality of the data to use with another classifier,
they can be used along with <a class="reference internal" href="generated/sklearn.feature_selection.SelectFromModel.html#sklearn.feature_selection.SelectFromModel" title="sklearn.feature_selection.SelectFromModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFromModel</span></code></a>
to select the non-zero coefficients. In particular, sparse estimators useful
for this purpose are 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> for regression, and
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> and <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a>
for classification:</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.svm</span> <span class="kn">import</span> <span class="n">LinearSVC</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.feature_selection</span> <span class="kn">import</span> <span class="n">SelectFromModel</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(150, 4)</span>
<span class="gp">>>> </span><span class="n">lsvc</span> <span class="o">=</span> <span class="n">LinearSVC</span><span class="p">(</span><span class="n">C</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">penalty</span><span class="o">=</span><span class="s2">"l1"</span><span class="p">,</span> <span class="n">dual</span><span class="o">=</span><span class="kc">False</span><span class="p">)</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="gp">>>> </span><span class="n">model</span> <span class="o">=</span> <span class="n">SelectFromModel</span><span class="p">(</span><span class="n">lsvc</span><span class="p">,</span> <span class="n">prefit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_new</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_new</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(150, 3)</span>
</pre></div>
</div>
<p>With SVMs and logistic-regression, the parameter C controls the sparsity:
the smaller C the fewer features selected. With Lasso, the higher the
alpha parameter, the fewer features selected.</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_dense_vs_sparse_data.html#sphx-glr-auto-examples-linear-model-plot-lasso-dense-vs-sparse-data-py"><span class="std std-ref">Lasso on dense and sparse data</span></a>.</p></li>
</ul>
</aside>
<p id="compressive-sensing"><details id="summary-anchor">
<summary class="btn btn-light">
<strong>L1-recovery and compressive sensing</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>For a good choice of alpha, the <a class="reference internal" href="linear_model.html#lasso"><span class="std std-ref">Lasso</span></a> can fully recover the
exact set of non-zero variables using only few observations, provided
certain specific conditions are met. In particular, the number of
samples should be “sufficiently large”, or L1 models will perform at
random, where “sufficiently large” depends on the number of non-zero
coefficients, the logarithm of the number of features, the amount of
noise, the smallest absolute value of non-zero coefficients, and the
structure of the design matrix X. In addition, the design matrix must
display certain specific properties, such as not being too correlated.</p>
<p>There is no general rule to select an alpha parameter for recovery of
non-zero coefficients. It can by set 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> or
<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>), though this may lead to
under-penalized models: including a small number of non-relevant variables
is not detrimental to prediction score. BIC
(<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>) tends, on the opposite, to set
high values of alpha.</p>
<aside class="topic">
<p class="topic-title">Reference</p>
<p>Richard G. Baraniuk “Compressive Sensing”, IEEE Signal
Processing Magazine [120] July 2007
<a class="reference external" href="http://users.isr.ist.utl.pt/~aguiar/CS_notes.pdf">http://users.isr.ist.utl.pt/~aguiar/CS_notes.pdf</a></p>
</aside>
<p></div>
</details></p>
</section>
<section id="tree-based-feature-selection">
<h3><span class="section-number">1.13.4.2. </span>Tree-based feature selection<a class="headerlink" href="#tree-based-feature-selection" title="Link to this heading">¶</a></h3>
<p>Tree-based estimators (see the <a class="reference internal" href="classes.html#module-sklearn.tree" title="sklearn.tree"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.tree</span></code></a> module and forest
of trees in the <a class="reference internal" href="classes.html#module-sklearn.ensemble" title="sklearn.ensemble"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.ensemble</span></code></a> module) can be used to compute
impurity-based feature importances, which in turn can be used to discard irrelevant
features (when coupled with the <a class="reference internal" href="generated/sklearn.feature_selection.SelectFromModel.html#sklearn.feature_selection.SelectFromModel" title="sklearn.feature_selection.SelectFromModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFromModel</span></code></a>
meta-transformer):</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.ensemble</span> <span class="kn">import</span> <span class="n">ExtraTreesClassifier</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.feature_selection</span> <span class="kn">import</span> <span class="n">SelectFromModel</span>
<span class="gp">>>> </span><span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(150, 4)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">ExtraTreesClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">clf</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="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">feature_importances_</span>
<span class="go">array([ 0.04..., 0.05..., 0.4..., 0.4...])</span>
<span class="gp">>>> </span><span class="n">model</span> <span class="o">=</span> <span class="n">SelectFromModel</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">prefit</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_new</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_new</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(150, 2)</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/ensemble/plot_forest_importances.html#sphx-glr-auto-examples-ensemble-plot-forest-importances-py"><span class="std std-ref">Feature importances with a forest of trees</span></a>: example on
synthetic data showing the recovery of the actually meaningful
features.</p></li>
<li><p><a class="reference internal" href="../auto_examples/ensemble/plot_forest_importances_faces.html#sphx-glr-auto-examples-ensemble-plot-forest-importances-faces-py"><span class="std std-ref">Pixel importances with a parallel forest of trees</span></a>: example
on face recognition data.</p></li>
</ul>
</aside>
</section>
</section>
<section id="sequential-feature-selection">
<span id="id3"></span><h2><span class="section-number">1.13.5. </span>Sequential Feature Selection<a class="headerlink" href="#sequential-feature-selection" title="Link to this heading">¶</a></h2>
<p>Sequential Feature Selection <a class="reference internal" href="#sfs" id="id4"><span>[sfs]</span></a> (SFS) is available in the
<a class="reference internal" href="generated/sklearn.feature_selection.SequentialFeatureSelector.html#sklearn.feature_selection.SequentialFeatureSelector" title="sklearn.feature_selection.SequentialFeatureSelector"><code class="xref py py-class docutils literal notranslate"><span class="pre">SequentialFeatureSelector</span></code></a> transformer.
SFS can be either forward or backward:</p>
<p>Forward-SFS is a greedy procedure that iteratively finds the best new feature
to add to the set of selected features. Concretely, we initially start with
zero features and find the one feature that maximizes a cross-validated score
when an estimator is trained on this single feature. Once that first feature
is selected, we repeat the procedure by adding a new feature to the set of
selected features. The procedure stops when the desired number of selected
features is reached, as determined by the <code class="docutils literal notranslate"><span class="pre">n_features_to_select</span></code> parameter.</p>
<p>Backward-SFS follows the same idea but works in the opposite direction:
instead of starting with no features and greedily adding features, we start
with <em>all</em> the features and greedily <em>remove</em> features from the set. The
<code class="docutils literal notranslate"><span class="pre">direction</span></code> parameter controls whether forward or backward SFS is used.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Detail on Sequential Feature Selection</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>In general, forward and backward selection do not yield equivalent results.
Also, one may be much faster than the other depending on the requested number
of selected features: if we have 10 features and ask for 7 selected features,
forward selection would need to perform 7 iterations while backward selection
would only need to perform 3.</p>
<p>SFS differs from <a class="reference internal" href="generated/sklearn.feature_selection.RFE.html#sklearn.feature_selection.RFE" title="sklearn.feature_selection.RFE"><code class="xref py py-class docutils literal notranslate"><span class="pre">RFE</span></code></a> and
<a class="reference internal" href="generated/sklearn.feature_selection.SelectFromModel.html#sklearn.feature_selection.SelectFromModel" title="sklearn.feature_selection.SelectFromModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFromModel</span></code></a> in that it does not
require the underlying model to expose a <code class="docutils literal notranslate"><span class="pre">coef_</span></code> or <code class="docutils literal notranslate"><span class="pre">feature_importances_</span></code>
attribute. It may however be slower considering that more models need to be
evaluated, compared to the other approaches. For example in backward
selection, the iteration going from <code class="docutils literal notranslate"><span class="pre">m</span></code> features to <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">-</span> <span class="pre">1</span></code> features using k-fold
cross-validation requires fitting <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">k</span></code> models, while
<a class="reference internal" href="generated/sklearn.feature_selection.RFE.html#sklearn.feature_selection.RFE" title="sklearn.feature_selection.RFE"><code class="xref py py-class docutils literal notranslate"><span class="pre">RFE</span></code></a> would require only a single fit, and
<a class="reference internal" href="generated/sklearn.feature_selection.SelectFromModel.html#sklearn.feature_selection.SelectFromModel" title="sklearn.feature_selection.SelectFromModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFromModel</span></code></a> always just does a single
fit and requires no iterations.</p>
<aside class="topic">
<p class="topic-title">Reference</p>
<div role="list" class="citation-list">
<div class="citation" id="sfs" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">sfs</a><span class="fn-bracket">]</span></span>
<p>Ferri et al, <a class="reference external" href="https://citeseerx.ist.psu.edu/doc_view/pid/5fedabbb3957bbb442802e012d829ee0629a01b6">Comparative study of techniques for
large-scale feature selection</a>.</p>
</div>
</div>
</aside>
<p></div>
</details></p>
<aside class="topic">
<p class="topic-title">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/feature_selection/plot_select_from_model_diabetes.html#sphx-glr-auto-examples-feature-selection-plot-select-from-model-diabetes-py"><span class="std std-ref">Model-based and sequential feature selection</span></a></p></li>
</ul>
</aside>
</section>
<section id="feature-selection-as-part-of-a-pipeline">
<h2><span class="section-number">1.13.6. </span>Feature selection as part of a pipeline<a class="headerlink" href="#feature-selection-as-part-of-a-pipeline" title="Link to this heading">¶</a></h2>
<p>Feature selection is usually used as a pre-processing step before doing
the actual learning. The recommended way to do this in scikit-learn is
to use a <a class="reference internal" href="generated/sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline"><code class="xref py py-class docutils literal notranslate"><span class="pre">Pipeline</span></code></a>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">clf</span> <span class="o">=</span> <span class="n">Pipeline</span><span class="p">([</span>
<span class="p">(</span><span class="s1">'feature_selection'</span><span class="p">,</span> <span class="n">SelectFromModel</span><span class="p">(</span><span class="n">LinearSVC</span><span class="p">(</span><span class="n">dual</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">penalty</span><span class="o">=</span><span class="s2">"l1"</span><span class="p">))),</span>
<span class="p">(</span><span class="s1">'classification'</span><span class="p">,</span> <span class="n">RandomForestClassifier</span><span class="p">())</span>
<span class="p">])</span>
<span class="n">clf</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>
</pre></div>
</div>
<p>In this snippet we make use of a <a class="reference internal" href="generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC" title="sklearn.svm.LinearSVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearSVC</span></code></a>
coupled with <a class="reference internal" href="generated/sklearn.feature_selection.SelectFromModel.html#sklearn.feature_selection.SelectFromModel" title="sklearn.feature_selection.SelectFromModel"><code class="xref py py-class docutils literal notranslate"><span class="pre">SelectFromModel</span></code></a>
to evaluate feature importances and select the most relevant features.
Then, a <a class="reference internal" href="generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier" title="sklearn.ensemble.RandomForestClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RandomForestClassifier</span></code></a> is trained on the
transformed output, i.e. using only relevant features. You can perform
similar operations with the other feature selection methods and also
classifiers that provide a way to evaluate feature importances of course.
See the <a class="reference internal" href="generated/sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline"><code class="xref py py-class docutils literal notranslate"><span class="pre">Pipeline</span></code></a> examples for more details.</p>
</section>
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