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<li><a class="reference internal" href="#">FastICA on 2D point clouds</a></li>
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<div class="section" id="fastica-on-2d-point-clouds">
<span id="example-plot-ica-vs-pca-py"></span><h1>FastICA on 2D point clouds<a class="headerlink" href="#fastica-on-2d-point-clouds" title="Permalink to this headline">ΒΆ</a></h1>
<p>Illustrate visually the results of <a class="reference internal" href="../modules/decompositions.html#ica"><em>Independent component analysis (ICA)</em></a> vs <a class="reference internal" href="../modules/decompositions.html#pca"><em>Principal component analysis (PCA)</em></a> in the
feature space.</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 accoutning 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 green 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>
<img alt="auto_examples/images/plot_ica_vs_pca.png" class="align-center" src="auto_examples/images/plot_ica_vs_pca.png" />
<p><strong>Python source code:</strong> <a class="reference download internal" href="../_downloads/plot_ica_vs_pca.py"><tt class="xref download docutils literal"><span class="pre">plot_ica_vs_pca.py</span></tt></a></p>
<div class="highlight-python"><div class="highlight"><pre><span class="k">print</span> <span class="n">__doc__</span>
<span class="c"># Authors: Alexandre Gramfort, Gael Varoquaux</span>
<span class="c"># License: BSD</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pylab</span> <span class="kn">as</span> <span class="nn">pl</span>
<span class="kn">from</span> <span class="nn">scikits.learn.pca</span> <span class="kn">import</span> <span class="n">PCA</span>
<span class="kn">from</span> <span class="nn">scikits.learn.fastica</span> <span class="kn">import</span> <span class="n">FastICA</span>
<span class="c">###############################################################################</span>
<span class="c"># Generate sample data</span>
<span class="n">S</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">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">2</span><span class="p">,</span> <span class="mi">10000</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.</span>
<span class="c"># Mix data</span>
<span class="n">A</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]</span> <span class="c"># Mixing matrix</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">A</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span> <span class="c"># Generate observations</span>
<span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><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="o">.</span><span class="n">T</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="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">ica</span> <span class="o">=</span> <span class="n">FastICA</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="c"># Estimate the sources</span>
<span class="n">S_ica_</span> <span class="o">/=</span> <span class="n">S_ica_</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">1</span><span class="p">)[:,</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
<span class="c">###############################################################################</span>
<span class="c"># Plot results</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="bp">None</span><span class="p">):</span>
<span class="n">pl</span><span class="o">.</span><span class="n">scatter</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="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="s">'o'</span><span class="p">,</span> <span class="n">linewidths</span><span class="o">=</span><span class="mi">0</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="k">if</span> <span class="n">axis_list</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span>
<span class="k">for</span> <span class="n">color</span><span class="p">,</span> <span class="n">axis</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">colors</span><span class="p">,</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>
<span class="c"># Trick to get legend to work</span>
<span class="n">pl</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mf">0.1</span><span class="o">*</span><span class="n">x_axis</span><span class="p">,</span> <span class="mf">0.1</span><span class="o">*</span><span class="n">y_axis</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</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="c"># pl.quiver(x_axis, y_axis, x_axis, y_axis, zorder=11, width=0.01,</span>
<span class="n">pl</span><span class="o">.</span><span class="n">quiver</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="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">pl</span><span class="o">.</span><span class="n">hlines</span><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>
<span class="n">pl</span><span class="o">.</span><span class="n">vlines</span><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>
<span class="n">pl</span><span class="o">.</span><span class="n">xlim</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>
<span class="n">pl</span><span class="o">.</span><span class="n">ylim</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>
<span class="n">pl</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'$x$'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'$y$'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">close</span><span class="p">(</span><span class="s">'all'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplot</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="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>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'True Independant 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="p">,</span> <span class="n">ica</span><span class="o">.</span><span class="n">get_mixing_matrix</span><span class="p">()]</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplot</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="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> <span class="n">np</span><span class="o">.</span><span class="n">std</span><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">pl</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s">'PCA'</span><span class="p">,</span> <span class="s">'ICA'</span><span class="p">],</span> <span class="n">loc</span><span class="o">=</span><span class="s">'upper left'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Observations'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplot</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="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> <span class="n">np</span><span class="o">.</span><span class="n">std</span><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">1</span><span class="p">)[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">])</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'PCA scores'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplot</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="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> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">S_ica_</span><span class="p">))</span>
<span class="n">pl</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'ICA estimated sources'</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">subplots_adjust</span><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">0.26</span><span class="p">)</span>
<span class="n">pl</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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