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<li><a class="reference internal" href="#">Unsupervised learning: seeking representations of the data</a><ul>
<li><a class="reference internal" href="#clustering-grouping-observations-together">Clustering: grouping observations together</a><ul>
<li><a class="reference internal" href="#k-means-clustering">K-means clustering</a></li>
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<section id="unsupervised-learning-seeking-representations-of-the-data">
<h1>Unsupervised learning: seeking representations of the data<a class="headerlink" href="#unsupervised-learning-seeking-representations-of-the-data" title="Link to this heading">¶</a></h1>
<section id="clustering-grouping-observations-together">
<h2>Clustering: grouping observations together<a class="headerlink" href="#clustering-grouping-observations-together" title="Link to this heading">¶</a></h2>
<aside class="topic">
<p class="topic-title">The problem solved in clustering</p>
<p>Given the iris dataset, if we knew that there were 3 types of iris, but
did not have access to a taxonomist to label them: we could try a
<strong>clustering task</strong>: split the observations into well-separated group
called <em>clusters</em>.</p>
</aside>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># Set the PRNG</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="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<section id="k-means-clustering">
<h3>K-means clustering<a class="headerlink" href="#k-means-clustering" title="Link to this heading">¶</a></h3>
<p>Note that there exist a lot of different clustering criteria and associated
algorithms. The simplest clustering algorithm is <a class="reference internal" href="../../modules/clustering.html#k-means"><span class="std std-ref">K-means</span></a>.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">cluster</span><span class="p">,</span> <span class="n">datasets</span>
<span class="gp">>>> </span><span class="n">X_iris</span><span class="p">,</span> <span class="n">y_iris</span> <span class="o">=</span> <span class="n">datasets</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">k_means</span> <span class="o">=</span> <span class="n">cluster</span><span class="o">.</span><span class="n">KMeans</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">k_means</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_iris</span><span class="p">)</span>
<span class="go">KMeans(n_clusters=3)</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">k_means</span><span class="o">.</span><span class="n">labels_</span><span class="p">[::</span><span class="mi">10</span><span class="p">])</span>
<span class="go">[1 1 1 1 1 2 0 0 0 0 2 2 2 2 2]</span>
<span class="gp">>>> </span><span class="nb">print</span><span class="p">(</span><span class="n">y_iris</span><span class="p">[::</span><span class="mi">10</span><span class="p">])</span>
<span class="go">[0 0 0 0 0 1 1 1 1 1 2 2 2 2 2]</span>
</pre></div>
</div>
<figure class="align-default">
<a class="reference external image-reference" href="../../auto_examples/cluster/plot_cluster_iris.html"><img alt="../../_images/sphx_glr_plot_cluster_iris_001.png" src="../../_images/sphx_glr_plot_cluster_iris_001.png" style="width: 630.0px; height: 504.0px;" /></a>
</figure>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>There is absolutely no guarantee of recovering a ground truth. First,
choosing the right number of clusters is hard. Second, the algorithm
is sensitive to initialization, and can fall into local minima,
although scikit-learn employs several tricks to mitigate this issue.</p>
<p>For instance, on the image above, we can observe the difference between the
ground-truth (bottom right figure) and different clustering. We do not
recover the expected labels, either because the number of cluster was
chosen to be to large (top left figure) or suffer from a bad initialization
(bottom left figure).</p>
<p><strong>It is therefore important to not over-interpret clustering results.</strong></p>
</div>
<aside class="topic">
<p class="topic-title"><strong>Application example: vector quantization</strong></p>
<p>Clustering in general and KMeans, in particular, can be seen as a way
of choosing a small number of exemplars to compress the information.
The problem is sometimes known as
<a class="reference external" href="https://en.wikipedia.org/wiki/Vector_quantization">vector quantization</a>.
For instance, this can be used to posterize an image:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sp</span>
<span class="gp">>>> </span><span class="k">try</span><span class="p">:</span>
<span class="gp">... </span> <span class="n">face</span> <span class="o">=</span> <span class="n">sp</span><span class="o">.</span><span class="n">face</span><span class="p">(</span><span class="n">gray</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">... </span><span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
<span class="gp">... </span> <span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">misc</span>
<span class="gp">... </span> <span class="n">face</span> <span class="o">=</span> <span class="n">misc</span><span class="o">.</span><span class="n">face</span><span class="p">(</span><span class="n">gray</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">face</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="c1"># We need an (n_sample, n_feature) array</span>
<span class="gp">>>> </span><span class="n">k_means</span> <span class="o">=</span> <span class="n">cluster</span><span class="o">.</span><span class="n">KMeans</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">n_init</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">k_means</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">KMeans(n_clusters=5, n_init=1)</span>
<span class="gp">>>> </span><span class="n">values</span> <span class="o">=</span> <span class="n">k_means</span><span class="o">.</span><span class="n">cluster_centers_</span><span class="o">.</span><span class="n">squeeze</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">labels</span> <span class="o">=</span> <span class="n">k_means</span><span class="o">.</span><span class="n">labels_</span>
<span class="gp">>>> </span><span class="n">face_compressed</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">choose</span><span class="p">(</span><span class="n">labels</span><span class="p">,</span> <span class="n">values</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">face_compressed</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</span> <span class="n">face</span><span class="o">.</span><span class="n">shape</span>
</pre></div>
</div>
</aside>
<p><strong>Raw image</strong></p>
<figure class="align-default">
<a class="reference external image-reference" href="../../auto_examples/cluster/plot_face_compress.html"><img alt="../../_images/sphx_glr_plot_face_compress_001.png" src="../../_images/sphx_glr_plot_face_compress_001.png" /></a>
</figure>
<p><strong>K-means quantization</strong></p>
<figure class="align-default">
<a class="reference external image-reference" href="../../auto_examples/cluster/plot_face_compress.html"><img alt="../../_images/sphx_glr_plot_face_compress_004.png" src="../../_images/sphx_glr_plot_face_compress_004.png" /></a>
</figure>
<p><strong>Equal bins</strong></p>
<figure class="align-default">
<a class="reference external image-reference" href="../../auto_examples/cluster/plot_face_compress.html"><img alt="../../_images/sphx_glr_plot_face_compress_002.png" src="../../_images/sphx_glr_plot_face_compress_002.png" /></a>
</figure>
</section>
<section id="hierarchical-agglomerative-clustering-ward">
<h3>Hierarchical agglomerative clustering: Ward<a class="headerlink" href="#hierarchical-agglomerative-clustering-ward" title="Link to this heading">¶</a></h3>
<p>A <a class="reference internal" href="../../modules/clustering.html#hierarchical-clustering"><span class="std std-ref">Hierarchical clustering</span></a> method is a type of cluster analysis
that aims to build a hierarchy of clusters. In general, the various approaches
of this technique are either:</p>
<ul class="simple">
<li><p><strong>Agglomerative</strong> - bottom-up approaches: each observation starts in its
own cluster, and clusters are iteratively merged in such a way to
minimize a <em>linkage</em> criterion. This approach is particularly interesting
when the clusters of interest are made of only a few observations. When
the number of clusters is large, it is much more computationally efficient
than k-means.</p></li>
<li><p><strong>Divisive</strong> - top-down approaches: all observations start in one
cluster, which is iteratively split as one moves down the hierarchy.
For estimating large numbers of clusters, this approach is both slow (due
to all observations starting as one cluster, which it splits recursively)
and statistically ill-posed.</p></li>
</ul>
<section id="connectivity-constrained-clustering">
<h4>Connectivity-constrained clustering<a class="headerlink" href="#connectivity-constrained-clustering" title="Link to this heading">¶</a></h4>
<p>With agglomerative clustering, it is possible to specify which samples can be
clustered together by giving a connectivity graph. Graphs in scikit-learn
are represented by their adjacency matrix. Often, a sparse matrix is used.
This can be useful, for instance, to retrieve connected regions (sometimes
also referred to as connected components) when clustering an image.</p>
<a class="reference external image-reference" href="../../auto_examples/cluster/plot_coin_ward_segmentation.html"><img alt="../../_images/sphx_glr_plot_coin_ward_segmentation_001.png" class="align-center" src="../../_images/sphx_glr_plot_coin_ward_segmentation_001.png" style="width: 200.0px; height: 200.0px;" /></a>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">skimage.data</span> <span class="kn">import</span> <span class="n">coins</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">scipy.ndimage</span> <span class="kn">import</span> <span class="n">gaussian_filter</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">skimage.transform</span> <span class="kn">import</span> <span class="n">rescale</span>
<span class="gp">>>> </span><span class="n">rescaled_coins</span> <span class="o">=</span> <span class="n">rescale</span><span class="p">(</span>
<span class="gp">... </span> <span class="n">gaussian_filter</span><span class="p">(</span><span class="n">coins</span><span class="p">(),</span> <span class="n">sigma</span><span class="o">=</span><span class="mi">2</span><span class="p">),</span>
<span class="gp">... </span> <span class="mf">0.2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s1">'reflect'</span><span class="p">,</span> <span class="n">anti_aliasing</span><span class="o">=</span><span class="kc">False</span>
<span class="gp">... </span><span class="p">)</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">reshape</span><span class="p">(</span><span class="n">rescaled_coins</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
</pre></div>
</div>
<p>We need a vectorized version of the image. <code class="docutils literal notranslate"><span class="pre">'rescaled_coins'</span></code> is a down-scaled
version of the coins image to speed up the process:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.feature_extraction</span> <span class="kn">import</span> <span class="n">grid_to_graph</span>
<span class="gp">>>> </span><span class="n">connectivity</span> <span class="o">=</span> <span class="n">grid_to_graph</span><span class="p">(</span><span class="o">*</span><span class="n">rescaled_coins</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
</pre></div>
</div>
<p>Define the graph structure of the data. Pixels connected to their neighbors:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">n_clusters</span> <span class="o">=</span> <span class="mi">27</span> <span class="c1"># number of regions</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">AgglomerativeClustering</span>
<span class="gp">>>> </span><span class="n">ward</span> <span class="o">=</span> <span class="n">AgglomerativeClustering</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="n">n_clusters</span><span class="p">,</span> <span class="n">linkage</span><span class="o">=</span><span class="s1">'ward'</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">connectivity</span><span class="o">=</span><span class="n">connectivity</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">ward</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">AgglomerativeClustering(connectivity=..., n_clusters=27)</span>
<span class="gp">>>> </span><span class="n">label</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">ward</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span> <span class="n">rescaled_coins</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="feature-agglomeration">
<h4>Feature agglomeration<a class="headerlink" href="#feature-agglomeration" title="Link to this heading">¶</a></h4>
<p>We have seen that sparsity could be used to mitigate the curse of
dimensionality, <em>i.e</em> an insufficient amount of observations compared to the
number of features. Another approach is to merge together similar
features: <strong>feature agglomeration</strong>. This approach can be implemented by
clustering in the feature direction, in other words clustering the
transposed data.</p>
<a class="reference external image-reference" href="../../auto_examples/cluster/plot_digits_agglomeration.html"><img alt="../../_images/sphx_glr_plot_digits_agglomeration_001.png" class="align-center" src="../../_images/sphx_glr_plot_digits_agglomeration_001.png" style="width: 227.99999999999997px; height: 199.49999999999997px;" /></a>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">digits</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_digits</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">images</span> <span class="o">=</span> <span class="n">digits</span><span class="o">.</span><span class="n">images</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">reshape</span><span class="p">(</span><span class="n">images</span><span class="p">,</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">images</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">connectivity</span> <span class="o">=</span> <span class="n">grid_to_graph</span><span class="p">(</span><span class="o">*</span><span class="n">images</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">agglo</span> <span class="o">=</span> <span class="n">cluster</span><span class="o">.</span><span class="n">FeatureAgglomeration</span><span class="p">(</span><span class="n">connectivity</span><span class="o">=</span><span class="n">connectivity</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">n_clusters</span><span class="o">=</span><span class="mi">32</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">agglo</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">FeatureAgglomeration(connectivity=..., n_clusters=32)</span>
<span class="gp">>>> </span><span class="n">X_reduced</span> <span class="o">=</span> <span class="n">agglo</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_approx</span> <span class="o">=</span> <span class="n">agglo</span><span class="o">.</span><span class="n">inverse_transform</span><span class="p">(</span><span class="n">X_reduced</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">images_approx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">X_approx</span><span class="p">,</span> <span class="n">images</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
</pre></div>
</div>
<aside class="topic">
<p class="topic-title"><code class="docutils literal notranslate"><span class="pre">transform</span></code> and <code class="docutils literal notranslate"><span class="pre">inverse_transform</span></code> methods</p>
<p>Some estimators expose a <code class="docutils literal notranslate"><span class="pre">transform</span></code> method, for instance to reduce
the dimensionality of the dataset.</p>
</aside>
</section>
</section>
</section>
<section id="decompositions-from-a-signal-to-components-and-loadings">
<h2>Decompositions: from a signal to components and loadings<a class="headerlink" href="#decompositions-from-a-signal-to-components-and-loadings" title="Link to this heading">¶</a></h2>
<aside class="topic">
<p class="topic-title"><strong>Components and loadings</strong></p>
<p>If X is our multivariate data, then the problem that we are trying to solve
is to rewrite it on a different observational basis: we want to learn
loadings L and a set of components C such that <em>X = L C</em>.
Different criteria exist to choose the components</p>
</aside>
<section id="principal-component-analysis-pca">
<h3>Principal component analysis: PCA<a class="headerlink" href="#principal-component-analysis-pca" title="Link to this heading">¶</a></h3>
<p><a class="reference internal" href="../../modules/decomposition.html#pca"><span class="std std-ref">Principal component analysis (PCA)</span></a> selects the successive components that explain the maximum variance in the
signal. Let’s create a synthetic 3-dimensional dataset.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># Create a signal with only 2 useful dimensions</span>
<span class="gp">>>> </span><span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">x3</span> <span class="o">=</span> <span class="n">x1</span> <span class="o">+</span> <span class="n">x2</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">concatenate</span><span class="p">([</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">x3</span><span class="p">],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>The point cloud spanned by the observations above is very flat in one
direction: one of the three univariate features (i.e. z-axis) can almost be exactly
computed using the other two.</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">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="gp">>>> </span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s1">'3d'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">])</span>
<span class="go"><...></span>
<span class="gp">>>> </span><span class="n">_</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlabel</span><span class="o">=</span><span class="s2">"x"</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">"y"</span><span class="p">,</span> <span class="n">zlabel</span><span class="o">=</span><span class="s2">"z"</span><span class="p">)</span>
</pre></div>
</div>
<figure class="align-center">
<img alt="../../_images/unsupervised_learning-1.png" class="plot-directive" src="../../_images/unsupervised_learning-1.png" />
</figure>
<p>PCA finds the directions in which the data is not <em>flat</em>.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">decomposition</span>
<span class="gp">>>> </span><span class="n">pca</span> <span class="o">=</span> <span class="n">decomposition</span><span class="o">.</span><span class="n">PCA</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()</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_</span><span class="p">)</span>
<span class="go">[ 2.18565811e+00 1.19346747e+00 8.43026679e-32]</span>
</pre></div>
</div>
<p>Looking at the explained variance, we see that only the first two components
are useful. PCA can be used to reduce dimensionality while preserving
most of the information. It will project the data on the principal subspace.</p>
<div class="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">set_params</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="go">PCA(n_components=2)</span>
<span class="gp">>>> </span><span class="n">X_reduced</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X_reduced</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(100, 2)</span>
</pre></div>
</div>
</section>
<section id="independent-component-analysis-ica">
<h3>Independent Component Analysis: ICA<a class="headerlink" href="#independent-component-analysis-ica" title="Link to this heading">¶</a></h3>
<p><a class="reference internal" href="../../modules/decomposition.html#ica"><span class="std std-ref">Independent component analysis (ICA)</span></a> selects components so that the distribution of their loadings carries
a maximum amount of independent information. It is able to recover
<strong>non-Gaussian</strong> independent signals:</p>
<a class="reference external image-reference" href="../../auto_examples/decomposition/plot_ica_blind_source_separation.html"><img alt="../../_images/sphx_glr_plot_ica_blind_source_separation_001.png" class="align-center" src="../../_images/sphx_glr_plot_ica_blind_source_separation_001.png" style="width: 448.0px; height: 336.0px;" /></a>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="c1"># Generate sample data</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">scipy</span> <span class="kn">import</span> <span class="n">signal</span>
<span class="gp">>>> </span><span class="n">time</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">2000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">s1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">time</span><span class="p">)</span> <span class="c1"># Signal 1 : sinusoidal signal</span>
<span class="gp">>>> </span><span class="n">s2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="n">time</span><span class="p">))</span> <span class="c1"># Signal 2 : square signal</span>
<span class="gp">>>> </span><span class="n">s3</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">sawtooth</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">time</span><span class="p">)</span> <span class="c1"># Signal 3: saw tooth signal</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">s3</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">+=</span> <span class="mf">0.2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="c1"># Add noise</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">/=</span> <span class="n">S</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="c1"># Standardize data</span>
<span class="gp">>>> </span><span class="c1"># Mix data</span>
<span class="gp">>>> </span><span class="n">A</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="mi">1</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="mf">0.5</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="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span> <span class="c1"># Mixing matrix</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">dot</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="c1"># Generate observations</span>
<span class="gp">>>> </span><span class="c1"># Compute ICA</span>
<span class="gp">>>> </span><span class="n">ica</span> <span class="o">=</span> <span class="n">decomposition</span><span class="o">.</span><span class="n">FastICA</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">S_</span> <span class="o">=</span> <span class="n">ica</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="c1"># Get the estimated sources</span>
<span class="gp">>>> </span><span class="n">A_</span> <span class="o">=</span> <span class="n">ica</span><span class="o">.</span><span class="n">mixing_</span><span class="o">.</span><span class="n">T</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">S_</span><span class="p">,</span> <span class="n">A_</span><span class="p">)</span> <span class="o">+</span> <span class="n">ica</span><span class="o">.</span><span class="n">mean_</span><span class="p">)</span>
<span class="go">True</span>
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