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<li><a class="reference internal" href="#">FastICA on 2D point clouds</a><ul>
<li><a class="reference internal" href="#generate-sample-data">Generate sample data</a></li>
<li><a class="reference internal" href="#plot-results">Plot results</a></li>
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<p><a class="reference internal" href="#sphx-glr-download-auto-examples-decomposition-plot-ica-vs-pca-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
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<section class="sphx-glr-example-title" id="fastica-on-2d-point-clouds">
<span id="sphx-glr-auto-examples-decomposition-plot-ica-vs-pca-py"></span><h1>FastICA on 2D point clouds<a class="headerlink" href="#fastica-on-2d-point-clouds" title="Link to this heading">¶</a></h1>
<p>This example illustrates visually in the feature space a comparison by
results using two different component analysis techniques.</p>
<p><a class="reference internal" href="../../modules/decomposition.html#ica"><span class="std std-ref">Independent component analysis (ICA)</span></a> vs <a class="reference internal" href="../../modules/decomposition.html#pca"><span class="std std-ref">Principal component analysis (PCA)</span></a>.</p>
<p>Representing ICA in the feature space gives the view of ‘geometric ICA’:
ICA is an algorithm that finds directions in the feature space
corresponding to projections with high non-Gaussianity. These directions
need not be orthogonal in the original feature space, but they are
orthogonal in the whitened feature space, in which all directions
correspond to the same variance.</p>
<p>PCA, on the other hand, finds orthogonal directions in the raw feature
space that correspond to directions accounting for maximum variance.</p>
<p>Here we simulate independent sources using a highly non-Gaussian
process, 2 student T with a low number of degrees of freedom (top left
figure). We mix them to create observations (top right figure).
In this raw observation space, directions identified by PCA are
represented by orange vectors. We represent the signal in the PCA space,
after whitening by the variance corresponding to the PCA vectors (lower
left). Running ICA corresponds to finding a rotation in this space to
identify the directions of largest non-Gaussianity (lower right).</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: Alexandre Gramfort, Gael Varoquaux</span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="generate-sample-data">
<h2>Generate sample data<a class="headerlink" href="#generate-sample-data" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">PCA</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA" title="sklearn.decomposition.FastICA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">FastICA</span></a>
<span class="n">rng</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/legacy.html#numpy.random.RandomState" title="numpy.random.RandomState" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span></a><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
<span class="n">S</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">standard_t</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">20000</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">S</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">*=</span> <span class="mf">2.0</span>
<span class="c1"># Mix data</span>
<span class="n">A</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span> <span class="c1"># Mixing matrix</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot" title="numpy.dot" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><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="n">pca</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">PCA</span></a><span class="p">()</span>
<span class="n">S_pca_</span> <span class="o">=</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="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">ica</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.decomposition.FastICA.html#sklearn.decomposition.FastICA" title="sklearn.decomposition.FastICA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">FastICA</span></a><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="n">rng</span><span class="p">,</span> <span class="n">whiten</span><span class="o">=</span><span class="s2">"arbitrary-variance"</span><span class="p">)</span>
<span class="n">S_ica_</span> <span class="o">=</span> <span class="n">ica</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="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="c1"># Estimate the sources</span>
</pre></div>
</div>
</section>
<section id="plot-results">
<h2>Plot results<a class="headerlink" href="#plot-results" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">def</span> <span class="nf">plot_samples</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">axis_list</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span>
<span class="n">S</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">S</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">"o"</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"steelblue"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">axis_list</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">for</span> <span class="n">axis</span><span class="p">,</span> <span class="n">color</span><span class="p">,</span> <span class="n">label</span> <span class="ow">in</span> <span class="n">axis_list</span><span class="p">:</span>
<span class="n">axis</span> <span class="o">/=</span> <span class="n">axis</span><span class="o">.</span><span class="n">std</span><span class="p">()</span>
<span class="n">x_axis</span><span class="p">,</span> <span class="n">y_axis</span> <span class="o">=</span> <span class="n">axis</span>
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<span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
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<span class="n">x_axis</span><span class="p">,</span>
<span class="n">y_axis</span><span class="p">,</span>
<span class="n">zorder</span><span class="o">=</span><span class="mi">11</span><span class="p">,</span>
<span class="n">width</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span>
<span class="n">scale</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.hlines.html#matplotlib.pyplot.hlines" title="matplotlib.pyplot.hlines" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">hlines</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
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<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="matplotlib.pyplot.xlabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"x"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="matplotlib.pyplot.ylabel" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"y"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">()</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</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="n">plot_samples</span><span class="p">(</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>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"True Independent Sources"</span><span class="p">)</span>
<span class="n">axis_list</span> <span class="o">=</span> <span class="p">[(</span><span class="n">pca</span><span class="o">.</span><span class="n">components_</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="s2">"orange"</span><span class="p">,</span> <span class="s2">"PCA"</span><span class="p">),</span> <span class="p">(</span><span class="n">ica</span><span class="o">.</span><span class="n">mixing_</span><span class="p">,</span> <span class="s2">"red"</span><span class="p">,</span> <span class="s2">"ICA"</span><span class="p">)]</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">plot_samples</span><span class="p">(</span><span class="n">X</span> <span class="o">/</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.std.html#numpy.std" title="numpy.std" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">std</span></a><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">axis_list</span><span class="o">=</span><span class="n">axis_list</span><span class="p">)</span>
<span class="n">legend</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="matplotlib.pyplot.legend" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>
<span class="n">legend</span><span class="o">.</span><span class="n">set_zorder</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Observations"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">plot_samples</span><span class="p">(</span><span class="n">S_pca_</span> <span class="o">/</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.std.html#numpy.std" title="numpy.std" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">std</span></a><span class="p">(</span><span class="n">S_pca_</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"PCA recovered signals"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">plot_samples</span><span class="p">(</span><span class="n">S_ica_</span> <span class="o">/</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.std.html#numpy.std" title="numpy.std" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">std</span></a><span class="p">(</span><span class="n">S_ica_</span><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"ICA recovered signals"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots_adjust.html#matplotlib.pyplot.subplots_adjust" title="matplotlib.pyplot.subplots_adjust" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span></a><span class="p">(</span><span class="mf">0.09</span><span class="p">,</span> <span class="mf">0.04</span><span class="p">,</span> <span class="mf">0.94</span><span class="p">,</span> <span class="mf">0.94</span><span class="p">,</span> <span class="mf">0.26</span><span class="p">,</span> <span class="mf">0.36</span><span class="p">)</span>
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