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<li class="toctree-l1 has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
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<li class="toctree-l2"><a class="reference internal" href="partial_dependence.html">5.1. Partial Dependence and Individual Conditional Expectation plots</a></li>
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<li class="toctree-l2"><a class="reference internal" href="compose.html">7.1. Pipelines and composite estimators</a></li>
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<li class="toctree-l2 current active"><a class="current reference internal" href="#">7.3. Preprocessing data</a></li>
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<li class="toctree-l2"><a class="reference internal" href="unsupervised_reduction.html">7.5. Unsupervised dimensionality reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="random_projection.html">7.6. Random Projection</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">7.7. Kernel Approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="metrics.html">7.8. Pairwise metrics, Affinities and Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">7.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../datasets/toy_dataset.html">8.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">8.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">8.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">8.4. Loading other datasets</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">9.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">9.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">9.3. Parallelism, resource management, and configuration</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">10. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">11. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">12. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="array_api.html">12.1. Array API support (experimental)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">13. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">14. External Resources, Videos and Talks</a></li>
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<section id="preprocessing-data">
<span id="preprocessing"></span><h1><span class="section-number">7.3. </span>Preprocessing data<a class="headerlink" href="#preprocessing-data" title="Link to this heading">#</a></h1>
<p>The <code class="docutils literal notranslate"><span class="pre">sklearn.preprocessing</span></code> package provides several common
utility functions and transformer classes to change raw feature vectors
into a representation that is more suitable for the downstream estimators.</p>
<p>In general, many learning algorithms such as linear models benefit from standardization of the data set
(see <a class="reference internal" href="../auto_examples/preprocessing/plot_scaling_importance.html#sphx-glr-auto-examples-preprocessing-plot-scaling-importance-py"><span class="std std-ref">Importance of Feature Scaling</span></a>).
If some outliers are present in the set, robust scalers or other transformers can
be more appropriate. The behaviors of the different scalers, transformers, and
normalizers on a dataset containing marginal outliers are highlighted in
<a class="reference internal" href="../auto_examples/preprocessing/plot_all_scaling.html#sphx-glr-auto-examples-preprocessing-plot-all-scaling-py"><span class="std std-ref">Compare the effect of different scalers on data with outliers</span></a>.</p>
<section id="standardization-or-mean-removal-and-variance-scaling">
<span id="preprocessing-scaler"></span><h2><span class="section-number">7.3.1. </span>Standardization, or mean removal and variance scaling<a class="headerlink" href="#standardization-or-mean-removal-and-variance-scaling" title="Link to this heading">#</a></h2>
<p><strong>Standardization</strong> of datasets is a <strong>common requirement for many
machine learning estimators</strong> implemented in scikit-learn; they might behave
badly if the individual features do not more or less look like standard
normally distributed data: Gaussian with <strong>zero mean and unit variance</strong>.</p>
<p>In practice we often ignore the shape of the distribution and just
transform the data to center it by removing the mean value of each
feature, then scale it by dividing non-constant features by their
standard deviation.</p>
<p>For instance, many elements used in the objective function of
a learning algorithm (such as the RBF kernel of Support Vector
Machines or the l1 and l2 regularizers of linear models) may assume that
all features are centered around zero or have variance in the same
order. If a feature has a variance that is orders of magnitude larger
than others, it might dominate the objective function and make the
estimator unable to learn from other features correctly as expected.</p>
<p>The <a class="reference internal" href="../api/sklearn.preprocessing.html#module-sklearn.preprocessing" title="sklearn.preprocessing"><code class="xref py py-mod docutils literal notranslate"><span class="pre">preprocessing</span></code></a> module provides the
<a class="reference internal" href="generated/sklearn.preprocessing.StandardScaler.html#sklearn.preprocessing.StandardScaler" title="sklearn.preprocessing.StandardScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">StandardScaler</span></code></a> utility class, which is a quick and
easy way to perform the following operation on an array-like
dataset:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">preprocessing</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">X_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mf">2.</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="gp">... </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="o">-</span><span class="mf">1.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">scaler</span> <span class="o">=</span> <span class="n">preprocessing</span><span class="o">.</span><span class="n">StandardScaler</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">scaler</span>
<span class="go">StandardScaler()</span>
<span class="gp">>>> </span><span class="n">scaler</span><span class="o">.</span><span class="n">mean_</span>
<span class="go">array([1. ..., 0. ..., 0.33...])</span>
<span class="gp">>>> </span><span class="n">scaler</span><span class="o">.</span><span class="n">scale_</span>
<span class="go">array([0.81..., 0.81..., 1.24...])</span>
<span class="gp">>>> </span><span class="n">X_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_scaled</span>
<span class="go">array([[ 0. ..., -1.22..., 1.33...],</span>
<span class="go"> [ 1.22..., 0. ..., -0.26...],</span>
<span class="go"> [-1.22..., 1.22..., -1.06...]])</span>
</pre></div>
</div>
<p>Scaled data has zero mean and unit variance:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X_scaled</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">array([0., 0., 0.])</span>
<span class="gp">>>> </span><span class="n">X_scaled</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">array([1., 1., 1.])</span>
</pre></div>
</div>
<p>This class implements the <code class="docutils literal notranslate"><span class="pre">Transformer</span></code> API to compute the mean and
standard deviation on a training set so as to be able to later re-apply the
same transformation on the testing set. This class is hence suitable for
use in the early steps of 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="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.datasets</span><span class="w"> </span><span class="kn">import</span> <span class="n">make_classification</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.linear_model</span><span class="w"> </span><span class="kn">import</span> <span class="n">LogisticRegression</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.model_selection</span><span class="w"> </span><span class="kn">import</span> <span class="n">train_test_split</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.pipeline</span><span class="w"> </span><span class="kn">import</span> <span class="n">make_pipeline</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.preprocessing</span><span class="w"> </span><span class="kn">import</span> <span class="n">StandardScaler</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">make_classification</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">pipe</span> <span class="o">=</span> <span class="n">make_pipeline</span><span class="p">(</span><span class="n">StandardScaler</span><span class="p">(),</span> <span class="n">LogisticRegression</span><span class="p">())</span>
<span class="gp">>>> </span><span class="n">pipe</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span> <span class="c1"># apply scaling on training data</span>
<span class="go">Pipeline(steps=[('standardscaler', StandardScaler()),</span>
<span class="go"> ('logisticregression', LogisticRegression())])</span>
<span class="gp">>>> </span><span class="n">pipe</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span><span class="p">)</span> <span class="c1"># apply scaling on testing data, without leaking training data.</span>
<span class="go">0.96</span>
</pre></div>
</div>
<p>It is possible to disable either centering or scaling by either
passing <code class="docutils literal notranslate"><span class="pre">with_mean=False</span></code> or <code class="docutils literal notranslate"><span class="pre">with_std=False</span></code> to the constructor
of <a class="reference internal" href="generated/sklearn.preprocessing.StandardScaler.html#sklearn.preprocessing.StandardScaler" title="sklearn.preprocessing.StandardScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">StandardScaler</span></code></a>.</p>
<section id="scaling-features-to-a-range">
<h3><span class="section-number">7.3.1.1. </span>Scaling features to a range<a class="headerlink" href="#scaling-features-to-a-range" title="Link to this heading">#</a></h3>
<p>An alternative standardization is scaling features to
lie between a given minimum and maximum value, often between zero and one,
or so that the maximum absolute value of each feature is scaled to unit size.
This can be achieved using <a class="reference internal" href="generated/sklearn.preprocessing.MinMaxScaler.html#sklearn.preprocessing.MinMaxScaler" title="sklearn.preprocessing.MinMaxScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">MinMaxScaler</span></code></a> or <a class="reference internal" href="generated/sklearn.preprocessing.MaxAbsScaler.html#sklearn.preprocessing.MaxAbsScaler" title="sklearn.preprocessing.MaxAbsScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">MaxAbsScaler</span></code></a>,
respectively.</p>
<p>The motivation to use this scaling includes robustness to very small
standard deviations of features and preserving zero entries in sparse data.</p>
<p>Here is an example to scale a toy data matrix to the <code class="docutils literal notranslate"><span class="pre">[0,</span> <span class="pre">1]</span></code> range:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mf">2.</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="gp">... </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="o">-</span><span class="mf">1.</span><span class="p">]])</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="n">min_max_scaler</span> <span class="o">=</span> <span class="n">preprocessing</span><span class="o">.</span><span class="n">MinMaxScaler</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">X_train_minmax</span> <span class="o">=</span> <span class="n">min_max_scaler</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train_minmax</span>
<span class="go">array([[0.5 , 0. , 1. ],</span>
<span class="go"> [1. , 0.5 , 0.33333333],</span>
<span class="go"> [0. , 1. , 0. ]])</span>
</pre></div>
</div>
<p>The same instance of the transformer can then be applied to some new test data
unseen during the fit call: the same scaling and shifting operations will be
applied to be consistent with the transformation performed on the train data:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mf">3.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">4.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">X_test_minmax</span> <span class="o">=</span> <span class="n">min_max_scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_test_minmax</span>
<span class="go">array([[-1.5 , 0. , 1.66666667]])</span>
</pre></div>
</div>
<p>It is possible to introspect the scaler attributes to find about the exact
nature of the transformation learned on the training data:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">min_max_scaler</span><span class="o">.</span><span class="n">scale_</span>
<span class="go">array([0.5 , 0.5 , 0.33...])</span>
<span class="gp">>>> </span><span class="n">min_max_scaler</span><span class="o">.</span><span class="n">min_</span>
<span class="go">array([0. , 0.5 , 0.33...])</span>
</pre></div>
</div>
<p>If <a class="reference internal" href="generated/sklearn.preprocessing.MinMaxScaler.html#sklearn.preprocessing.MinMaxScaler" title="sklearn.preprocessing.MinMaxScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">MinMaxScaler</span></code></a> is given an explicit <code class="docutils literal notranslate"><span class="pre">feature_range=(min,</span> <span class="pre">max)</span></code> the
full formula is:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">X_std</span> <span class="o">=</span> <span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">X</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">-</span> <span class="n">X</span><span class="o">.</span><span class="n">min</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">))</span>
<span class="n">X_scaled</span> <span class="o">=</span> <span class="n">X_std</span> <span class="o">*</span> <span class="p">(</span><span class="nb">max</span> <span class="o">-</span> <span class="nb">min</span><span class="p">)</span> <span class="o">+</span> <span class="nb">min</span>
</pre></div>
</div>
<p><a class="reference internal" href="generated/sklearn.preprocessing.MaxAbsScaler.html#sklearn.preprocessing.MaxAbsScaler" title="sklearn.preprocessing.MaxAbsScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">MaxAbsScaler</span></code></a> works in a very similar fashion, but scales in a way
that the training data lies within the range <code class="docutils literal notranslate"><span class="pre">[-1,</span> <span class="pre">1]</span></code> by dividing through
the largest maximum value in each feature. It is meant for data
that is already centered at zero or sparse data.</p>
<p>Here is how to use the toy data from the previous example with this scaler:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">X_train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span> <span class="mf">2.</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="gp">... </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="o">-</span><span class="mf">1.</span><span class="p">]])</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="n">max_abs_scaler</span> <span class="o">=</span> <span class="n">preprocessing</span><span class="o">.</span><span class="n">MaxAbsScaler</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">X_train_maxabs</span> <span class="o">=</span> <span class="n">max_abs_scaler</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train_maxabs</span>
<span class="go">array([[ 0.5, -1. , 1. ],</span>
<span class="go"> [ 1. , 0. , 0. ],</span>
<span class="go"> [ 0. , 1. , -0.5]])</span>
<span class="gp">>>> </span><span class="n">X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span> <span class="o">-</span><span class="mf">3.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">4.</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">X_test_maxabs</span> <span class="o">=</span> <span class="n">max_abs_scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_test_maxabs</span>
<span class="go">array([[-1.5, -1. , 2. ]])</span>
<span class="gp">>>> </span><span class="n">max_abs_scaler</span><span class="o">.</span><span class="n">scale_</span>
<span class="go">array([2., 1., 2.])</span>
</pre></div>
</div>
</section>
<section id="scaling-sparse-data">
<h3><span class="section-number">7.3.1.2. </span>Scaling sparse data<a class="headerlink" href="#scaling-sparse-data" title="Link to this heading">#</a></h3>
<p>Centering sparse data would destroy the sparseness structure in the data, and
thus rarely is a sensible thing to do. However, it can make sense to scale
sparse inputs, especially if features are on different scales.</p>
<p><a class="reference internal" href="generated/sklearn.preprocessing.MaxAbsScaler.html#sklearn.preprocessing.MaxAbsScaler" title="sklearn.preprocessing.MaxAbsScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">MaxAbsScaler</span></code></a> was specifically designed for scaling
sparse data, and is the recommended way to go about this.
However, <a class="reference internal" href="generated/sklearn.preprocessing.StandardScaler.html#sklearn.preprocessing.StandardScaler" title="sklearn.preprocessing.StandardScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">StandardScaler</span></code></a> can accept <code class="docutils literal notranslate"><span class="pre">scipy.sparse</span></code>
matrices as input, as long as <code class="docutils literal notranslate"><span class="pre">with_mean=False</span></code> is explicitly passed
to the constructor. Otherwise a <code class="docutils literal notranslate"><span class="pre">ValueError</span></code> will be raised as
silently centering would break the sparsity and would often crash the
execution by allocating excessive amounts of memory unintentionally.
<a class="reference internal" href="generated/sklearn.preprocessing.RobustScaler.html#sklearn.preprocessing.RobustScaler" title="sklearn.preprocessing.RobustScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">RobustScaler</span></code></a> cannot be fitted to sparse inputs, but you can use
the <code class="docutils literal notranslate"><span class="pre">transform</span></code> method on sparse inputs.</p>
<p>Note that the scalers accept both Compressed Sparse Rows and Compressed
Sparse Columns format (see <code class="docutils literal notranslate"><span class="pre">scipy.sparse.csr_matrix</span></code> and
<code class="docutils literal notranslate"><span class="pre">scipy.sparse.csc_matrix</span></code>). Any other sparse input will be <strong>converted to
the Compressed Sparse Rows representation</strong>. To avoid unnecessary memory
copies, it is recommended to choose the CSR or CSC representation upstream.</p>
<p>Finally, if the centered data is expected to be small enough, explicitly
converting the input to an array using the <code class="docutils literal notranslate"><span class="pre">toarray</span></code> method of sparse matrices
is another option.</p>
</section>
<section id="scaling-data-with-outliers">
<h3><span class="section-number">7.3.1.3. </span>Scaling data with outliers<a class="headerlink" href="#scaling-data-with-outliers" title="Link to this heading">#</a></h3>
<p>If your data contains many outliers, scaling using the mean and variance
of the data is likely to not work very well. In these cases, you can use
<a class="reference internal" href="generated/sklearn.preprocessing.RobustScaler.html#sklearn.preprocessing.RobustScaler" title="sklearn.preprocessing.RobustScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">RobustScaler</span></code></a> as a drop-in replacement instead. It uses
more robust estimates for the center and range of your data.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">Further discussion on the importance of centering and scaling data is
available on this FAQ: <a class="reference external" href="http://www.faqs.org/faqs/ai-faq/neural-nets/part2/section-16.html">Should I normalize/standardize/rescale the data?</a></p>
</div>
</details><details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="scaling-vs-whitening">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Scaling vs Whitening<a class="headerlink" href="#scaling-vs-whitening" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">It is sometimes not enough to center and scale the features
independently, since a downstream model can further make some assumption
on the linear independence of the features.</p>
<p class="sd-card-text">To address this issue you can use <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> with
<code class="docutils literal notranslate"><span class="pre">whiten=True</span></code> to further remove the linear correlation across features.</p>
</div>
</details></section>
<section id="centering-kernel-matrices">
<span id="kernel-centering"></span><h3><span class="section-number">7.3.1.4. </span>Centering kernel matrices<a class="headerlink" href="#centering-kernel-matrices" title="Link to this heading">#</a></h3>
<p>If you have a kernel matrix of a kernel <span class="math notranslate nohighlight">\(K\)</span> that computes a dot product
in a feature space (possibly implicitly) defined by a function
<span class="math notranslate nohighlight">\(\phi(\cdot)\)</span>, a <a class="reference internal" href="generated/sklearn.preprocessing.KernelCenterer.html#sklearn.preprocessing.KernelCenterer" title="sklearn.preprocessing.KernelCenterer"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelCenterer</span></code></a> can transform the kernel matrix
so that it contains inner products in the feature space defined by <span class="math notranslate nohighlight">\(\phi\)</span>
followed by the removal of the mean in that space. In other words,
<a class="reference internal" href="generated/sklearn.preprocessing.KernelCenterer.html#sklearn.preprocessing.KernelCenterer" title="sklearn.preprocessing.KernelCenterer"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelCenterer</span></code></a> computes the centered Gram matrix associated to a
positive semidefinite kernel <span class="math notranslate nohighlight">\(K\)</span>.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="mathematical-formulation">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">We can have a look at the mathematical formulation now that we have the
intuition. Let <span class="math notranslate nohighlight">\(K\)</span> be a kernel matrix of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_samples)</span></code>
computed from <span class="math notranslate nohighlight">\(X\)</span>, a data matrix of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_features)</span></code>,
during the <code class="docutils literal notranslate"><span class="pre">fit</span></code> step. <span class="math notranslate nohighlight">\(K\)</span> is defined by</p>
<div class="math notranslate nohighlight">
\[K(X, X) = \phi(X) . \phi(X)^{T}\]</div>
<p class="sd-card-text"><span class="math notranslate nohighlight">\(\phi(X)\)</span> is a function mapping of <span class="math notranslate nohighlight">\(X\)</span> to a Hilbert space. A
centered kernel <span class="math notranslate nohighlight">\(\tilde{K}\)</span> is defined as:</p>
<div class="math notranslate nohighlight">
\[\tilde{K}(X, X) = \tilde{\phi}(X) . \tilde{\phi}(X)^{T}\]</div>
<p class="sd-card-text">where <span class="math notranslate nohighlight">\(\tilde{\phi}(X)\)</span> results from centering <span class="math notranslate nohighlight">\(\phi(X)\)</span> in the
Hilbert space.</p>
<p class="sd-card-text">Thus, one could compute <span class="math notranslate nohighlight">\(\tilde{K}\)</span> by mapping <span class="math notranslate nohighlight">\(X\)</span> using the
function <span class="math notranslate nohighlight">\(\phi(\cdot)\)</span> and center the data in this new space. However,
kernels are often used because they allow some algebra calculations that
avoid computing explicitly this mapping using <span class="math notranslate nohighlight">\(\phi(\cdot)\)</span>. Indeed, one
can implicitly center as shown in Appendix B in <a class="reference internal" href="#scholkopf1998" id="id1"><span>[Scholkopf1998]</span></a>:</p>
<div class="math notranslate nohighlight">
\[\tilde{K} = K - 1_{\text{n}_{samples}} K - K 1_{\text{n}_{samples}} + 1_{\text{n}_{samples}} K 1_{\text{n}_{samples}}\]</div>
<p class="sd-card-text"><span class="math notranslate nohighlight">\(1_{\text{n}_{samples}}\)</span> is a matrix of <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_samples)</span></code> where
all entries are equal to <span class="math notranslate nohighlight">\(\frac{1}{\text{n}_{samples}}\)</span>. In the
<code class="docutils literal notranslate"><span class="pre">transform</span></code> step, the kernel becomes <span class="math notranslate nohighlight">\(K_{test}(X, Y)\)</span> defined as:</p>
<div class="math notranslate nohighlight">
\[K_{test}(X, Y) = \phi(Y) . \phi(X)^{T}\]</div>
<p class="sd-card-text"><span class="math notranslate nohighlight">\(Y\)</span> is the test dataset of shape <code class="docutils literal notranslate"><span class="pre">(n_samples_test,</span> <span class="pre">n_features)</span></code> and thus
<span class="math notranslate nohighlight">\(K_{test}\)</span> is of shape <code class="docutils literal notranslate"><span class="pre">(n_samples_test,</span> <span class="pre">n_samples)</span></code>. In this case,
centering <span class="math notranslate nohighlight">\(K_{test}\)</span> is done as:</p>
<div class="math notranslate nohighlight">
\[\tilde{K}_{test}(X, Y) = K_{test} - 1'_{\text{n}_{samples}} K - K_{test} 1_{\text{n}_{samples}} + 1'_{\text{n}_{samples}} K 1_{\text{n}_{samples}}\]</div>
<p class="sd-card-text"><span class="math notranslate nohighlight">\(1'_{\text{n}_{samples}}\)</span> is a matrix of shape
<code class="docutils literal notranslate"><span class="pre">(n_samples_test,</span> <span class="pre">n_samples)</span></code> where all entries are equal to
<span class="math notranslate nohighlight">\(\frac{1}{\text{n}_{samples}}\)</span>.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="scholkopf1998" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Scholkopf1998</a><span class="fn-bracket">]</span></span>
<p class="sd-card-text">B. Schölkopf, A. Smola, and K.R. Müller,
<a class="reference external" href="https://www.mlpack.org/papers/kpca.pdf">“Nonlinear component analysis as a kernel eigenvalue problem.”</a>
Neural computation 10.5 (1998): 1299-1319.</p>
</div>
</div>
</div>
</details></section>
</section>
<section id="non-linear-transformation">
<span id="preprocessing-transformer"></span><h2><span class="section-number">7.3.2. </span>Non-linear transformation<a class="headerlink" href="#non-linear-transformation" title="Link to this heading">#</a></h2>
<p>Two types of transformations are available: quantile transforms and power
transforms. Both quantile and power transforms are based on monotonic
transformations of the features and thus preserve the rank of the values
along each feature.</p>
<p>Quantile transforms put all features into the same desired distribution based
on the formula <span class="math notranslate nohighlight">\(G^{-1}(F(X))\)</span> where <span class="math notranslate nohighlight">\(F\)</span> is the cumulative
distribution function of the feature and <span class="math notranslate nohighlight">\(G^{-1}\)</span> the
<a class="reference external" href="https://en.wikipedia.org/wiki/Quantile_function">quantile function</a> of the
desired output distribution <span class="math notranslate nohighlight">\(G\)</span>. This formula is using the two following
facts: (i) if <span class="math notranslate nohighlight">\(X\)</span> is a random variable with a continuous cumulative
distribution function <span class="math notranslate nohighlight">\(F\)</span> then <span class="math notranslate nohighlight">\(F(X)\)</span> is uniformly distributed on
<span class="math notranslate nohighlight">\([0,1]\)</span>; (ii) if <span class="math notranslate nohighlight">\(U\)</span> is a random variable with uniform distribution
on <span class="math notranslate nohighlight">\([0,1]\)</span> then <span class="math notranslate nohighlight">\(G^{-1}(U)\)</span> has distribution <span class="math notranslate nohighlight">\(G\)</span>. By performing
a rank transformation, a quantile transform smooths out unusual distributions
and is less influenced by outliers than scaling methods. It does, however,
distort correlations and distances within and across features.</p>
<p>Power transforms are a family of parametric transformations that aim to map
data from any distribution to as close to a Gaussian distribution.</p>
<section id="mapping-to-a-uniform-distribution">
<h3><span class="section-number">7.3.2.1. </span>Mapping to a Uniform distribution<a class="headerlink" href="#mapping-to-a-uniform-distribution" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.preprocessing.QuantileTransformer.html#sklearn.preprocessing.QuantileTransformer" title="sklearn.preprocessing.QuantileTransformer"><code class="xref py py-class docutils literal notranslate"><span class="pre">QuantileTransformer</span></code></a> provides a non-parametric
transformation to map the data to a uniform distribution
with values between 0 and 1:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.datasets</span><span class="w"> </span><span class="kn">import</span> <span class="n">load_iris</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.model_selection</span><span class="w"> </span><span class="kn">import</span> <span class="n">train_test_split</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_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">quantile_transformer</span> <span class="o">=</span> <span class="n">preprocessing</span><span class="o">.</span><span class="n">QuantileTransformer</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_train_trans</span> <span class="o">=</span> <span class="n">quantile_transformer</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_test_trans</span> <span class="o">=</span> <span class="n">quantile_transformer</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">X_train</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">25</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="go">array([ 4.3, 5.1, 5.8, 6.5, 7.9])</span>
</pre></div>
</div>
<p>This feature corresponds to the sepal length in cm. Once the quantile
transformation is applied, those landmarks approach closely the percentiles
previously defined:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">X_train_trans</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">25</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="gp">... </span>
<span class="go">array([ 0.00... , 0.24..., 0.49..., 0.73..., 0.99... ])</span>
</pre></div>
</div>
<p>This can be confirmed on an independent testing set with similar remarks:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">X_test</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">25</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="gp">... </span>
<span class="go">array([ 4.4 , 5.125, 5.75 , 6.175, 7.3 ])</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">percentile</span><span class="p">(</span><span class="n">X_test_trans</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">25</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">75</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
<span class="gp">... </span>