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<section id="pca">
<h1>PCA<a class="headerlink" href="#pca" title="Link to this heading">#</a></h1>
<dl class="py class">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">sklearn.decomposition.</span></span><span class="sig-name descname"><span class="pre">PCA</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n_components</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">copy</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">whiten</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">svd_solver</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'auto'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">iterated_power</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'auto'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_oversamples</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">power_iteration_normalizer</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'auto'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">random_state</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_pca.py#L115"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA" title="Link to this definition">#</a></dt>
<dd><p>Principal component analysis (PCA).</p>
<p>Linear dimensionality reduction using Singular Value Decomposition of the
data to project it to a lower dimensional space. The input data is centered
but not scaled for each feature before applying the SVD.</p>
<p>It uses the LAPACK implementation of the full SVD or a randomized truncated
SVD by the method of Halko et al. 2009, depending on the shape of the input
data and the number of components to extract.</p>
<p>With sparse inputs, the ARPACK implementation of the truncated SVD can be
used (i.e. through <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.svds.html#scipy.sparse.linalg.svds" title="(in SciPy v1.14.1)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.sparse.linalg.svds</span></code></a>). Alternatively, one
may consider <a class="reference internal" href="sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-class docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a> where the data are not centered.</p>
<p>Notice that this class only supports sparse inputs for some solvers such as
“arpack” and “covariance_eigh”. See <a class="reference internal" href="sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-class docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a> for an
alternative with sparse data.</p>
<p>For a usage example, see
<a class="reference internal" href="../../auto_examples/decomposition/plot_pca_iris.html#sphx-glr-auto-examples-decomposition-plot-pca-iris-py"><span class="std std-ref">Principal Component Analysis (PCA) on Iris Dataset</span></a></p>
<p>Read more in the <a class="reference internal" href="../decomposition.html#pca"><span class="std std-ref">User Guide</span></a>.</p>
<dl class="field-list">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl>
<dt><strong>n_components</strong><span class="classifier">int, float or ‘mle’, default=None</span></dt><dd><p>Number of components to keep.
if n_components is not set all components are kept:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n_components</span> <span class="o">==</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span>
</pre></div>
</div>
<p>If <code class="docutils literal notranslate"><span class="pre">n_components</span> <span class="pre">==</span> <span class="pre">'mle'</span></code> and <code class="docutils literal notranslate"><span class="pre">svd_solver</span> <span class="pre">==</span> <span class="pre">'full'</span></code>, Minka’s
MLE is used to guess the dimension. Use of <code class="docutils literal notranslate"><span class="pre">n_components</span> <span class="pre">==</span> <span class="pre">'mle'</span></code>
will interpret <code class="docutils literal notranslate"><span class="pre">svd_solver</span> <span class="pre">==</span> <span class="pre">'auto'</span></code> as <code class="docutils literal notranslate"><span class="pre">svd_solver</span> <span class="pre">==</span> <span class="pre">'full'</span></code>.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">n_components</span> <span class="pre"><</span> <span class="pre">1</span></code> and <code class="docutils literal notranslate"><span class="pre">svd_solver</span> <span class="pre">==</span> <span class="pre">'full'</span></code>, select the
number of components such that the amount of variance that needs to be
explained is greater than the percentage specified by n_components.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">svd_solver</span> <span class="pre">==</span> <span class="pre">'arpack'</span></code>, the number of components must be
strictly less than the minimum of n_features and n_samples.</p>
<p>Hence, the None case results in:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n_components</span> <span class="o">==</span> <span class="nb">min</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
</pre></div>
</div>
</dd>
<dt><strong>copy</strong><span class="classifier">bool, default=True</span></dt><dd><p>If False, data passed to fit are overwritten and running
fit(X).transform(X) will not yield the expected results,
use fit_transform(X) instead.</p>
</dd>
<dt><strong>whiten</strong><span class="classifier">bool, default=False</span></dt><dd><p>When True (False by default) the <code class="docutils literal notranslate"><span class="pre">components_</span></code> vectors are multiplied
by the square root of n_samples and then divided by the singular values
to ensure uncorrelated outputs with unit component-wise variances.</p>
<p>Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometime
improve the predictive accuracy of the downstream estimators by
making their data respect some hard-wired assumptions.</p>
</dd>
<dt><strong>svd_solver</strong><span class="classifier">{‘auto’, ‘full’, ‘covariance_eigh’, ‘arpack’, ‘randomized’}, default=’auto’</span></dt><dd><dl class="simple">
<dt>“auto” :</dt><dd><p>The solver is selected by a default ‘auto’ policy is based on <code class="docutils literal notranslate"><span class="pre">X.shape</span></code> and
<code class="docutils literal notranslate"><span class="pre">n_components</span></code>: if the input data has fewer than 1000 features and
more than 10 times as many samples, then the “covariance_eigh”
solver is used. Otherwise, if the input data is larger than 500x500
and the number of components to extract is lower than 80% of the
smallest dimension of the data, then the more efficient
“randomized” method is selected. Otherwise the exact “full” SVD is
computed and optionally truncated afterwards.</p>
</dd>
<dt>“full” :</dt><dd><p>Run exact full SVD calling the standard LAPACK solver via
<code class="docutils literal notranslate"><span class="pre">scipy.linalg.svd</span></code> and select the components by postprocessing</p>
</dd>
<dt>“covariance_eigh” :</dt><dd><p>Precompute the covariance matrix (on centered data), run a
classical eigenvalue decomposition on the covariance matrix
typically using LAPACK and select the components by postprocessing.
This solver is very efficient for n_samples >> n_features and small
n_features. It is, however, not tractable otherwise for large
n_features (large memory footprint required to materialize the
covariance matrix). Also note that compared to the “full” solver,
this solver effectively doubles the condition number and is
therefore less numerical stable (e.g. on input data with a large
range of singular values).</p>
</dd>
<dt>“arpack” :</dt><dd><p>Run SVD truncated to <code class="docutils literal notranslate"><span class="pre">n_components</span></code> calling ARPACK solver via
<code class="docutils literal notranslate"><span class="pre">scipy.sparse.linalg.svds</span></code>. It requires strictly
<code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><</span> <span class="pre">n_components</span> <span class="pre"><</span> <span class="pre">min(X.shape)</span></code></p>
</dd>
<dt>“randomized” :</dt><dd><p>Run randomized SVD by the method of Halko et al.</p>
</dd>
</dl>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.18.0.</span></p>
</div>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 1.5: </span>Added the ‘covariance_eigh’ solver.</p>
</div>
</dd>
<dt><strong>tol</strong><span class="classifier">float, default=0.0</span></dt><dd><p>Tolerance for singular values computed by svd_solver == ‘arpack’.
Must be of range [0.0, infinity).</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.18.0.</span></p>
</div>
</dd>
<dt><strong>iterated_power</strong><span class="classifier">int or ‘auto’, default=’auto’</span></dt><dd><p>Number of iterations for the power method computed by
svd_solver == ‘randomized’.
Must be of range [0, infinity).</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.18.0.</span></p>
</div>
</dd>
<dt><strong>n_oversamples</strong><span class="classifier">int, default=10</span></dt><dd><p>This parameter is only relevant when <code class="docutils literal notranslate"><span class="pre">svd_solver="randomized"</span></code>.
It corresponds to the additional number of random vectors to sample the
range of <code class="docutils literal notranslate"><span class="pre">X</span></code> so as to ensure proper conditioning. See
<a class="reference internal" href="sklearn.utils.extmath.randomized_svd.html#sklearn.utils.extmath.randomized_svd" title="sklearn.utils.extmath.randomized_svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">randomized_svd</span></code></a> for more details.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 1.1.</span></p>
</div>
</dd>
<dt><strong>power_iteration_normalizer</strong><span class="classifier">{‘auto’, ‘QR’, ‘LU’, ‘none’}, default=’auto’</span></dt><dd><p>Power iteration normalizer for randomized SVD solver.
Not used by ARPACK. See <a class="reference internal" href="sklearn.utils.extmath.randomized_svd.html#sklearn.utils.extmath.randomized_svd" title="sklearn.utils.extmath.randomized_svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">randomized_svd</span></code></a>
for more details.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 1.1.</span></p>
</div>
</dd>
<dt><strong>random_state</strong><span class="classifier">int, RandomState instance or None, default=None</span></dt><dd><p>Used when the ‘arpack’ or ‘randomized’ solvers are used. Pass an int
for reproducible results across multiple function calls.
See <a class="reference internal" href="../../glossary.html#term-random_state"><span class="xref std std-term">Glossary</span></a>.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.18.0.</span></p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Attributes<span class="colon">:</span></dt>
<dd class="field-even"><dl>
<dt><strong>components_</strong><span class="classifier">ndarray of shape (n_components, n_features)</span></dt><dd><p>Principal axes in feature space, representing the directions of
maximum variance in the data. Equivalently, the right singular
vectors of the centered input data, parallel to its eigenvectors.
The components are sorted by decreasing <code class="docutils literal notranslate"><span class="pre">explained_variance_</span></code>.</p>
</dd>
<dt><strong>explained_variance_</strong><span class="classifier">ndarray of shape (n_components,)</span></dt><dd><p>The amount of variance explained by each of the selected components.
The variance estimation uses <code class="docutils literal notranslate"><span class="pre">n_samples</span> <span class="pre">-</span> <span class="pre">1</span></code> degrees of freedom.</p>
<p>Equal to n_components largest eigenvalues
of the covariance matrix of X.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.18.</span></p>
</div>
</dd>
<dt><strong>explained_variance_ratio_</strong><span class="classifier">ndarray of shape (n_components,)</span></dt><dd><p>Percentage of variance explained by each of the selected components.</p>
<p>If <code class="docutils literal notranslate"><span class="pre">n_components</span></code> is not set then all components are stored and the
sum of the ratios is equal to 1.0.</p>
</dd>
<dt><strong>singular_values_</strong><span class="classifier">ndarray of shape (n_components,)</span></dt><dd><p>The singular values corresponding to each of the selected components.
The singular values are equal to the 2-norms of the <code class="docutils literal notranslate"><span class="pre">n_components</span></code>
variables in the lower-dimensional space.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.19.</span></p>
</div>
</dd>
<dt><strong>mean_</strong><span class="classifier">ndarray of shape (n_features,)</span></dt><dd><p>Per-feature empirical mean, estimated from the training set.</p>
<p>Equal to <code class="docutils literal notranslate"><span class="pre">X.mean(axis=0)</span></code>.</p>
</dd>
<dt><strong>n_components_</strong><span class="classifier">int</span></dt><dd><p>The estimated number of components. When n_components is set
to ‘mle’ or a number between 0 and 1 (with svd_solver == ‘full’) this
number is estimated from input data. Otherwise it equals the parameter
n_components, or the lesser value of n_features and n_samples
if n_components is None.</p>
</dd>
<dt><strong>n_samples_</strong><span class="classifier">int</span></dt><dd><p>Number of samples in the training data.</p>
</dd>
<dt><strong>noise_variance_</strong><span class="classifier">float</span></dt><dd><p>The estimated noise covariance following the Probabilistic PCA model
from Tipping and Bishop 1999. See “Pattern Recognition and
Machine Learning” by C. Bishop, 12.2.1 p. 574 or
<a class="reference external" href="http://www.miketipping.com/papers/met-mppca.pdf">http://www.miketipping.com/papers/met-mppca.pdf</a>. It is required to
compute the estimated data covariance and score samples.</p>
<p>Equal to the average of (min(n_features, n_samples) - n_components)
smallest eigenvalues of the covariance matrix of X.</p>
</dd>
<dt><strong>n_features_in_</strong><span class="classifier">int</span></dt><dd><p>Number of features seen during <a class="reference internal" href="../../glossary.html#term-fit"><span class="xref std std-term">fit</span></a>.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 0.24.</span></p>
</div>
</dd>
<dt><strong>feature_names_in_</strong><span class="classifier">ndarray of shape (<code class="docutils literal notranslate"><span class="pre">n_features_in_</span></code>,)</span></dt><dd><p>Names of features seen during <a class="reference internal" href="../../glossary.html#term-fit"><span class="xref std std-term">fit</span></a>. Defined only when <code class="docutils literal notranslate"><span class="pre">X</span></code>
has feature names that are all strings.</p>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 1.0.</span></p>
</div>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-obj docutils literal notranslate"><span class="pre">KernelPCA</span></code></a></dt><dd><p>Kernel Principal Component Analysis.</p>
</dd>
<dt><a class="reference internal" href="sklearn.decomposition.SparsePCA.html#sklearn.decomposition.SparsePCA" title="sklearn.decomposition.SparsePCA"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SparsePCA</span></code></a></dt><dd><p>Sparse Principal Component Analysis.</p>
</dd>
<dt><a class="reference internal" href="sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-obj docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a></dt><dd><p>Dimensionality reduction using truncated SVD.</p>
</dd>
<dt><a class="reference internal" href="sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-obj docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a></dt><dd><p>Incremental Principal Component Analysis.</p>
</dd>
</dl>
</div>
<p class="rubric">References</p>
<p>For n_components == ‘mle’, this class uses the method from:
<a class="reference external" href="https://tminka.github.io/papers/pca/minka-pca.pdf">Minka, T. P.. “Automatic choice of dimensionality for PCA”.
In NIPS, pp. 598-604</a></p>
<p>Implements the probabilistic PCA model from:
<a class="reference external" href="http://www.miketipping.com/papers/met-mppca.pdf">Tipping, M. E., and Bishop, C. M. (1999). “Probabilistic principal
component analysis”. Journal of the Royal Statistical Society:
Series B (Statistical Methodology), 61(3), 611-622.</a>
via the score and score_samples methods.</p>
<p>For svd_solver == ‘arpack’, refer to <code class="docutils literal notranslate"><span class="pre">scipy.sparse.linalg.svds</span></code>.</p>
<p>For svd_solver == ‘randomized’, see:
<a class="reference external" href="https://doi.org/10.1137/090771806">Halko, N., Martinsson, P. G., and Tropp, J. A. (2011).
“Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions”.
SIAM review, 53(2), 217-288.</a>
and also
<a class="reference external" href="https://doi.org/10.1016/j.acha.2010.02.003">Martinsson, P. G., Rokhlin, V., and Tygert, M. (2011).
“A randomized algorithm for the decomposition of matrices”.
Applied and Computational Harmonic Analysis, 30(1), 47-68.</a></p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span>
<span class="gp">>>> </span><span class="n">X</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="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</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">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">pca</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="go">PCA(n_components=2)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">explained_variance_ratio_</span><span class="p">)</span>
<span class="go">[0.9924... 0.0075...]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">singular_values_</span><span class="p">)</span>
<span class="go">[6.30061... 0.54980...]</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">svd_solver</span><span class="o">=</span><span class="s1">'full'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">pca</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="go">PCA(n_components=2, svd_solver='full')</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">explained_variance_ratio_</span><span class="p">)</span>
<span class="go">[0.9924... 0.00755...]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">singular_values_</span><span class="p">)</span>
<span class="go">[6.30061... 0.54980...]</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">svd_solver</span><span class="o">=</span><span class="s1">'arpack'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">pca</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="go">PCA(n_components=1, svd_solver='arpack')</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">explained_variance_ratio_</span><span class="p">)</span>
<span class="go">[0.99244...]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">singular_values_</span><span class="p">)</span>
<span class="go">[6.30061...]</span>
</pre></div>
</div>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.fit">
<span class="sig-name descname"><span class="pre">fit</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_pca.py#L424"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.fit" title="Link to this definition">#</a></dt>
<dd><p>Fit the model with X.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">{array-like, sparse matrix} of shape (n_samples, n_features)</span></dt><dd><p>Training data, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> is the number of samples
and <code class="docutils literal notranslate"><span class="pre">n_features</span></code> is the number of features.</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Ignored.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>self</strong><span class="classifier">object</span></dt><dd><p>Returns the instance itself.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.fit_transform">
<span class="sig-name descname"><span class="pre">fit_transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_pca.py#L445"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.fit_transform" title="Link to this definition">#</a></dt>
<dd><p>Fit the model with X and apply the dimensionality reduction on X.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">{array-like, sparse matrix} of shape (n_samples, n_features)</span></dt><dd><p>Training data, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> is the number of samples
and <code class="docutils literal notranslate"><span class="pre">n_features</span></code> is the number of features.</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Ignored.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>X_new</strong><span class="classifier">ndarray of shape (n_samples, n_components)</span></dt><dd><p>Transformed values.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>This method returns a Fortran-ordered array. To convert it to a
C-ordered array, use ‘np.ascontiguousarray’.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.get_covariance">
<span class="sig-name descname"><span class="pre">get_covariance</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_base.py#L25"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.get_covariance" title="Link to this definition">#</a></dt>
<dd><p>Compute data covariance with the generative model.</p>
<p><code class="docutils literal notranslate"><span class="pre">cov</span> <span class="pre">=</span> <span class="pre">components_.T</span> <span class="pre">*</span> <span class="pre">S**2</span> <span class="pre">*</span> <span class="pre">components_</span> <span class="pre">+</span> <span class="pre">sigma2</span> <span class="pre">*</span> <span class="pre">eye(n_features)</span></code>
where S**2 contains the explained variances, and sigma2 contains the
noise variances.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>cov</strong><span class="classifier">array of shape=(n_features, n_features)</span></dt><dd><p>Estimated covariance of data.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.get_feature_names_out">
<span class="sig-name descname"><span class="pre">get_feature_names_out</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input_features</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/base.py#L898"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.get_feature_names_out" title="Link to this definition">#</a></dt>
<dd><p>Get output feature names for transformation.</p>
<p>The feature names out will prefixed by the lowercased class name. For
example, if the transformer outputs 3 features, then the feature names
out are: <code class="docutils literal notranslate"><span class="pre">["class_name0",</span> <span class="pre">"class_name1",</span> <span class="pre">"class_name2"]</span></code>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>input_features</strong><span class="classifier">array-like of str or None, default=None</span></dt><dd><p>Only used to validate feature names with the names seen in <code class="docutils literal notranslate"><span class="pre">fit</span></code>.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>feature_names_out</strong><span class="classifier">ndarray of str objects</span></dt><dd><p>Transformed feature names.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.get_metadata_routing">
<span class="sig-name descname"><span class="pre">get_metadata_routing</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/utils/_metadata_requests.py#L1497"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.get_metadata_routing" title="Link to this definition">#</a></dt>
<dd><p>Get metadata routing of this object.</p>
<p>Please check <a class="reference internal" href="../../metadata_routing.html#metadata-routing"><span class="std std-ref">User Guide</span></a> on how the routing
mechanism works.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>routing</strong><span class="classifier">MetadataRequest</span></dt><dd><p>A <a class="reference internal" href="sklearn.utils.metadata_routing.MetadataRequest.html#sklearn.utils.metadata_routing.MetadataRequest" title="sklearn.utils.metadata_routing.MetadataRequest"><code class="xref py py-class docutils literal notranslate"><span class="pre">MetadataRequest</span></code></a> encapsulating
routing information.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.get_params">
<span class="sig-name descname"><span class="pre">get_params</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">deep</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/base.py#L221"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.get_params" title="Link to this definition">#</a></dt>
<dd><p>Get parameters for this estimator.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>deep</strong><span class="classifier">bool, default=True</span></dt><dd><p>If True, will return the parameters for this estimator and
contained subobjects that are estimators.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>params</strong><span class="classifier">dict</span></dt><dd><p>Parameter names mapped to their values.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.get_precision">
<span class="sig-name descname"><span class="pre">get_precision</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_base.py#L53"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.get_precision" title="Link to this definition">#</a></dt>
<dd><p>Compute data precision matrix with the generative model.</p>
<p>Equals the inverse of the covariance but computed with
the matrix inversion lemma for efficiency.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>precision</strong><span class="classifier">array, shape=(n_features, n_features)</span></dt><dd><p>Estimated precision of data.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.inverse_transform">
<span class="sig-name descname"><span class="pre">inverse_transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_base.py#L167"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.inverse_transform" title="Link to this definition">#</a></dt>
<dd><p>Transform data back to its original space.</p>
<p>In other words, return an input <code class="docutils literal notranslate"><span class="pre">X_original</span></code> whose transform would be X.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like of shape (n_samples, n_components)</span></dt><dd><p>New data, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> is the number of samples
and <code class="docutils literal notranslate"><span class="pre">n_components</span></code> is the number of components.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt>X_original array-like of shape (n_samples, n_features)</dt><dd><p>Original data, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> is the number of samples
and <code class="docutils literal notranslate"><span class="pre">n_features</span></code> is the number of features.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>If whitening is enabled, inverse_transform will compute the
exact inverse operation, which includes reversing whitening.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.score">
<span class="sig-name descname"><span class="pre">score</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_pca.py#L827"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.score" title="Link to this definition">#</a></dt>
<dd><p>Return the average log-likelihood of all samples.</p>
<p>See. “Pattern Recognition and Machine Learning”
by C. Bishop, 12.2.1 p. 574
or <a class="reference external" href="http://www.miketipping.com/papers/met-mppca.pdf">http://www.miketipping.com/papers/met-mppca.pdf</a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like of shape (n_samples, n_features)</span></dt><dd><p>The data.</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Ignored.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>ll</strong><span class="classifier">float</span></dt><dd><p>Average log-likelihood of the samples under the current model.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.score_samples">
<span class="sig-name descname"><span class="pre">score_samples</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_pca.py#L800"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.score_samples" title="Link to this definition">#</a></dt>
<dd><p>Return the log-likelihood of each sample.</p>
<p>See. “Pattern Recognition and Machine Learning”
by C. Bishop, 12.2.1 p. 574
or <a class="reference external" href="http://www.miketipping.com/papers/met-mppca.pdf">http://www.miketipping.com/papers/met-mppca.pdf</a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like of shape (n_samples, n_features)</span></dt><dd><p>The data.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>ll</strong><span class="classifier">ndarray of shape (n_samples,)</span></dt><dd><p>Log-likelihood of each sample under the current model.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.set_output">
<span class="sig-name descname"><span class="pre">set_output</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">transform</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/utils/_set_output.py#L392"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.set_output" title="Link to this definition">#</a></dt>
<dd><p>Set output container.</p>
<p>See <a class="reference internal" href="../../auto_examples/miscellaneous/plot_set_output.html#sphx-glr-auto-examples-miscellaneous-plot-set-output-py"><span class="std std-ref">Introducing the set_output API</span></a>
for an example on how to use the API.</p>
<dl class="field-list">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl>
<dt><strong>transform</strong><span class="classifier">{“default”, “pandas”, “polars”}, default=None</span></dt><dd><p>Configure output of <code class="docutils literal notranslate"><span class="pre">transform</span></code> and <code class="docutils literal notranslate"><span class="pre">fit_transform</span></code>.</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">"default"</span></code>: Default output format of a transformer</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"pandas"</span></code>: DataFrame output</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">"polars"</span></code>: Polars output</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">None</span></code>: Transform configuration is unchanged</p></li>
</ul>
<div class="versionadded">
<p><span class="versionmodified added">Added in version 1.4: </span><code class="docutils literal notranslate"><span class="pre">"polars"</span></code> option was added.</p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>self</strong><span class="classifier">estimator instance</span></dt><dd><p>Estimator instance.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.set_params">
<span class="sig-name descname"><span class="pre">set_params</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">params</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/base.py#L245"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.set_params" title="Link to this definition">#</a></dt>
<dd><p>Set the parameters of this estimator.</p>
<p>The method works on simple estimators as well as on nested objects
(such as <a class="reference internal" href="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>). The latter have
parameters of the form <code class="docutils literal notranslate"><span class="pre"><component>__<parameter></span></code> so that it’s
possible to update each component of a nested object.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>**params</strong><span class="classifier">dict</span></dt><dd><p>Estimator parameters.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>self</strong><span class="classifier">estimator instance</span></dt><dd><p>Estimator instance.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="sklearn.decomposition.PCA.transform">
<span class="sig-name descname"><span class="pre">transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">X</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/4ee3afa55/sklearn/decomposition/_base.py#L116"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.decomposition.PCA.transform" title="Link to this definition">#</a></dt>
<dd><p>Apply dimensionality reduction to X.</p>
<p>X is projected on the first principal components previously extracted
from a training set.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">{array-like, sparse matrix} of shape (n_samples, n_features)</span></dt><dd><p>New data, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> is the number of samples
and <code class="docutils literal notranslate"><span class="pre">n_features</span></code> is the number of features.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>X_new</strong><span class="classifier">array-like of shape (n_samples, n_components)</span></dt><dd><p>Projection of X in the first principal components, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>
is the number of samples and <code class="docutils literal notranslate"><span class="pre">n_components</span></code> is the number of the components.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
</dd></dl>
<section id="gallery-examples">
<h2>Gallery examples<a class="headerlink" href="#gallery-examples" title="Link to this heading">#</a></h2>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="We are pleased to announce the release of scikit-learn 1.5! Many bug fixes and improvements were added, as well as some key new features. Below we detail the highlights of this release. For an exhaustive list of all the changes, please refer to the release notes <release_notes_1_5>."><img alt="" src="../../_images/sphx_glr_plot_release_highlights_1_5_0_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/release_highlights/plot_release_highlights_1_5_0.html#sphx-glr-auto-examples-release-highlights-plot-release-highlights-1-5-0-py"><span class="std std-ref">Release Highlights for scikit-learn 1.5</span></a></p>
<div class="sphx-glr-thumbnail-title">Release Highlights for scikit-learn 1.5</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="We are pleased to announce the release of scikit-learn 1.4! Many bug fixes and improvements were added, as well as some new key features. We detail below a few of the major features of this release. For an exhaustive list of all the changes, please refer to the release notes <release_notes_1_4>."><img alt="" src="../../_images/sphx_glr_plot_release_highlights_1_4_0_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/release_highlights/plot_release_highlights_1_4_0.html#sphx-glr-auto-examples-release-highlights-plot-release-highlights-1-4-0-py"><span class="std std-ref">Release Highlights for scikit-learn 1.4</span></a></p>
<div class="sphx-glr-thumbnail-title">Release Highlights for scikit-learn 1.4</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="In this example we compare the various initialization strategies for K-means in terms of runtime and quality of the results."><img alt="" src="../../_images/sphx_glr_plot_kmeans_digits_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/cluster/plot_kmeans_digits.html#sphx-glr-auto-examples-cluster-plot-kmeans-digits-py"><span class="std std-ref">A demo of K-Means clustering on the handwritten digits data</span></a></p>
<div class="sphx-glr-thumbnail-title">A demo of K-Means clustering on the handwritten digits data</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example compares Principal Component Regression (PCR) and Partial Least Squares Regression (PLS) on a toy dataset. Our goal is to illustrate how PLS can outperform PCR when the target is strongly correlated with some directions in the data that have a low variance."><img alt="" src="../../_images/sphx_glr_plot_pcr_vs_pls_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/cross_decomposition/plot_pcr_vs_pls.html#sphx-glr-auto-examples-cross-decomposition-plot-pcr-vs-pls-py"><span class="std std-ref">Principal Component Regression vs Partial Least Squares Regression</span></a></p>
<div class="sphx-glr-thumbnail-title">Principal Component Regression vs Partial Least Squares Regression</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="An example of estimating sources from noisy data."><img alt="" src="../../_images/sphx_glr_plot_ica_blind_source_separation_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_ica_blind_source_separation.html#sphx-glr-auto-examples-decomposition-plot-ica-blind-source-separation-py"><span class="std std-ref">Blind source separation using FastICA</span></a></p>
<div class="sphx-glr-thumbnail-title">Blind source separation using FastICA</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The Iris dataset represents 3 kind of Iris flowers (Setosa, Versicolour and Virginica) with 4 attributes: sepal length, sepal width, petal length and petal width."><img alt="" src="../../_images/sphx_glr_plot_pca_vs_lda_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_pca_vs_lda.html#sphx-glr-auto-examples-decomposition-plot-pca-vs-lda-py"><span class="std std-ref">Comparison of LDA and PCA 2D projection of Iris dataset</span></a></p>
<div class="sphx-glr-thumbnail-title">Comparison of LDA and PCA 2D projection of Iris dataset</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example applies to olivetti_faces_dataset different unsupervised matrix decomposition (dimension reduction) methods from the module sklearn.decomposition (see the documentation chapter decompositions)."><img alt="" src="../../_images/sphx_glr_plot_faces_decomposition_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_faces_decomposition.html#sphx-glr-auto-examples-decomposition-plot-faces-decomposition-py"><span class="std std-ref">Faces dataset decompositions</span></a></p>
<div class="sphx-glr-thumbnail-title">Faces dataset decompositions</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Investigating the Iris dataset, we see that sepal length, petal length and petal width are highly correlated. Sepal width is less redundant. Matrix decomposition techniques can uncover these latent patterns. Applying rotations to the resulting components does not inherently improve the predictive value of the derived latent space, but can help visualise their structure; here, for example, the varimax rotation, which is found by maximizing the squared variances of the weights, finds a structure where the second component only loads positively on sepal width."><img alt="" src="../../_images/sphx_glr_plot_varimax_fa_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_varimax_fa.html#sphx-glr-auto-examples-decomposition-plot-varimax-fa-py"><span class="std std-ref">Factor Analysis (with rotation) to visualize patterns</span></a></p>
<div class="sphx-glr-thumbnail-title">Factor Analysis (with rotation) to visualize patterns</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates visually in the feature space a comparison by results using two different component analysis techniques."><img alt="" src="../../_images/sphx_glr_plot_ica_vs_pca_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_ica_vs_pca.html#sphx-glr-auto-examples-decomposition-plot-ica-vs-pca-py"><span class="std std-ref">FastICA on 2D point clouds</span></a></p>
<div class="sphx-glr-thumbnail-title">FastICA on 2D point clouds</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Incremental principal component analysis (IPCA) is typically used as a replacement for principal component analysis (PCA) when the dataset to be decomposed is too large to fit in memory. IPCA builds a low-rank approximation for the input data using an amount of memory which is independent of the number of input data samples. It is still dependent on the input data features, but changing the batch size allows for control of memory usage."><img alt="" src="../../_images/sphx_glr_plot_incremental_pca_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_incremental_pca.html#sphx-glr-auto-examples-decomposition-plot-incremental-pca-py"><span class="std std-ref">Incremental PCA</span></a></p>
<div class="sphx-glr-thumbnail-title">Incremental PCA</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows the difference between the Principal Components Analysis (~sklearn.decomposition.PCA) and its kernelized version (~sklearn.decomposition.KernelPCA)."><img alt="" src="../../_images/sphx_glr_plot_kernel_pca_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_kernel_pca.html#sphx-glr-auto-examples-decomposition-plot-kernel-pca-py"><span class="std std-ref">Kernel PCA</span></a></p>
<div class="sphx-glr-thumbnail-title">Kernel PCA</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Probabilistic PCA and Factor Analysis are probabilistic models. The consequence is that the likelihood of new data can be used for model selection and covariance estimation. Here we compare PCA and FA with cross-validation on low rank data corrupted with homoscedastic noise (noise variance is the same for each feature) or heteroscedastic noise (noise variance is the different for each feature). In a second step we compare the model likelihood to the likelihoods obtained from shrinkage covariance estimators."><img alt="" src="../../_images/sphx_glr_plot_pca_vs_fa_model_selection_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html#sphx-glr-auto-examples-decomposition-plot-pca-vs-fa-model-selection-py"><span class="std std-ref">Model selection with Probabilistic PCA and Factor Analysis (FA)</span></a></p>
<div class="sphx-glr-thumbnail-title">Model selection with Probabilistic PCA and Factor Analysis (FA)</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows a well known decomposition technique known as Principal Component Analysis (PCA) on the Iris dataset."><img alt="" src="../../_images/sphx_glr_plot_pca_iris_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/decomposition/plot_pca_iris.html#sphx-glr-auto-examples-decomposition-plot-pca-iris-py"><span class="std std-ref">Principal Component Analysis (PCA) on Iris Dataset</span></a></p>
<div class="sphx-glr-thumbnail-title">Principal Component Analysis (PCA) on Iris Dataset</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The dataset used in this example is a preprocessed excerpt of the "Labeled Faces in the Wild", aka LFW_: http://vis-www.cs.umass.edu/lfw/lfw-funneled.tgz (233MB)"><img alt="" src="../../_images/sphx_glr_plot_face_recognition_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/applications/plot_face_recognition.html#sphx-glr-auto-examples-applications-plot-face-recognition-py"><span class="std std-ref">Faces recognition example using eigenfaces and SVMs</span></a></p>
<div class="sphx-glr-thumbnail-title">Faces recognition example using eigenfaces and SVMs</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows how to use KernelPCA to denoise images. In short, we take advantage of the approximation function learned during fit to reconstruct the original image."><img alt="" src="../../_images/sphx_glr_plot_digits_denoising_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/applications/plot_digits_denoising.html#sphx-glr-auto-examples-applications-plot-digits-denoising-py"><span class="std std-ref">Image denoising using kernel PCA</span></a></p>
<div class="sphx-glr-thumbnail-title">Image denoising using kernel PCA</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="An illustration of the metric and non-metric MDS on generated noisy data."><img alt="" src="../../_images/sphx_glr_plot_mds_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/manifold/plot_mds.html#sphx-glr-auto-examples-manifold-plot-mds-py"><span class="std std-ref">Multi-dimensional scaling</span></a></p>
<div class="sphx-glr-thumbnail-title">Multi-dimensional scaling</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The default configuration for displaying a pipeline in a Jupyter Notebook is 'diagram' where set_config(display='diagram'). To deactivate HTML representation, use set_config(display='text')."><img alt="" src="../../_images/sphx_glr_plot_pipeline_display_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/miscellaneous/plot_pipeline_display.html#sphx-glr-auto-examples-miscellaneous-plot-pipeline-display-py"><span class="std std-ref">Displaying Pipelines</span></a></p>
<div class="sphx-glr-thumbnail-title">Displaying Pipelines</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="An example illustrating the approximation of the feature map of an RBF kernel."><img alt="" src="../../_images/sphx_glr_plot_kernel_approximation_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/miscellaneous/plot_kernel_approximation.html#sphx-glr-auto-examples-miscellaneous-plot-kernel-approximation-py"><span class="std std-ref">Explicit feature map approximation for RBF kernels</span></a></p>
<div class="sphx-glr-thumbnail-title">Explicit feature map approximation for RBF kernels</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example simulates a multi-label document classification problem. The dataset is generated randomly based on the following process:"><img alt="" src="../../_images/sphx_glr_plot_multilabel_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/miscellaneous/plot_multilabel.html#sphx-glr-auto-examples-miscellaneous-plot-multilabel-py"><span class="std std-ref">Multilabel classification</span></a></p>
<div class="sphx-glr-thumbnail-title">Multilabel classification</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example balances model complexity and cross-validated score by finding a decent accuracy within 1 standard deviation of the best accuracy score while minimising the number of PCA components [1]."><img alt="" src="../../_images/sphx_glr_plot_grid_search_refit_callable_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/model_selection/plot_grid_search_refit_callable.html#sphx-glr-auto-examples-model-selection-plot-grid-search-refit-callable-py"><span class="std std-ref">Balance model complexity and cross-validated score</span></a></p>
<div class="sphx-glr-thumbnail-title">Balance model complexity and cross-validated score</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Sample usage of Neighborhood Components Analysis for dimensionality reduction."><img alt="" src="../../_images/sphx_glr_plot_nca_dim_reduction_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/neighbors/plot_nca_dim_reduction.html#sphx-glr-auto-examples-neighbors-plot-nca-dim-reduction-py"><span class="std std-ref">Dimensionality Reduction with Neighborhood Components Analysis</span></a></p>
<div class="sphx-glr-thumbnail-title">Dimensionality Reduction with Neighborhood Components Analysis</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows how kernel density estimation (KDE), a powerful non-parametric density estimation technique, can be used to learn a generative model for a dataset. With this generative model in place, new samples can be drawn. These new samples reflect the underlying model of the data."><img alt="" src="../../_images/sphx_glr_plot_digits_kde_sampling_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/neighbors/plot_digits_kde_sampling.html#sphx-glr-auto-examples-neighbors-plot-digits-kde-sampling-py"><span class="std std-ref">Kernel Density Estimation</span></a></p>
<div class="sphx-glr-thumbnail-title">Kernel Density Estimation</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Datasets can often contain components that require different feature extraction and processing pipelines. This scenario might occur when:"><img alt="" src="../../_images/sphx_glr_plot_column_transformer_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/compose/plot_column_transformer.html#sphx-glr-auto-examples-compose-plot-column-transformer-py"><span class="std std-ref">Column Transformer with Heterogeneous Data Sources</span></a></p>
<div class="sphx-glr-thumbnail-title">Column Transformer with Heterogeneous Data Sources</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="In many real-world examples, there are many ways to extract features from a dataset. Often it is beneficial to combine several methods to obtain good performance. This example shows how to use FeatureUnion to combine features obtained by PCA and univariate selection."><img alt="" src="../../_images/sphx_glr_plot_feature_union_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/compose/plot_feature_union.html#sphx-glr-auto-examples-compose-plot-feature-union-py"><span class="std std-ref">Concatenating multiple feature extraction methods</span></a></p>
<div class="sphx-glr-thumbnail-title">Concatenating multiple feature extraction methods</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The PCA does an unsupervised dimensionality reduction, while the logistic regression does the prediction."><img alt="" src="../../_images/sphx_glr_plot_digits_pipe_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/compose/plot_digits_pipe.html#sphx-glr-auto-examples-compose-plot-digits-pipe-py"><span class="std std-ref">Pipelining: chaining a PCA and a logistic regression</span></a></p>
<div class="sphx-glr-thumbnail-title">Pipelining: chaining a PCA and a logistic regression</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example constructs a pipeline that does dimensionality reduction followed by prediction with a support vector classifier. It demonstrates the use of GridSearchCV and Pipeline to optimize over different classes of estimators in a single CV run -- unsupervised PCA and NMF dimensionality reductions are compared to univariate feature selection during the grid search."><img alt="" src="../../_images/sphx_glr_plot_compare_reduction_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/compose/plot_compare_reduction.html#sphx-glr-auto-examples-compose-plot-compare-reduction-py"><span class="std std-ref">Selecting dimensionality reduction with Pipeline and GridSearchCV</span></a></p>
<div class="sphx-glr-thumbnail-title">Selecting dimensionality reduction with Pipeline and GridSearchCV</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Feature scaling through standardization, also called Z-score normalization, is an important preprocessing step for many machine learning algorithms. It involves rescaling each feature such that it has a standard deviation of 1 and a mean of 0."><img alt="" src="../../_images/sphx_glr_plot_scaling_importance_thumb.png" />
<p><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></p>
<div class="sphx-glr-thumbnail-title">Importance of Feature Scaling</div>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sklearn.decomposition.PCA"><code class="docutils literal notranslate"><span class="pre">PCA</span></code></a><ul class="nav section-nav flex-column visible">
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