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<li><a class="reference internal" href="#">Principal components analysis (PCA)</a><ul>
<li><a class="reference internal" href="#create-the-data">Create the data</a></li>
<li><a class="reference internal" href="#plot-the-figures">Plot the figures</a></li>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-decomposition-plot-pca-3d-py"><span class="std std-ref">here</span></a>
to download the full example code or to run this example in your browser via Binder</p>
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<section class="sphx-glr-example-title" id="principal-components-analysis-pca">
<span id="sphx-glr-auto-examples-decomposition-plot-pca-3d-py"></span><h1>Principal components analysis (PCA)<a class="headerlink" href="#principal-components-analysis-pca" title="Permalink to this heading">¶</a></h1>
<p>These figures aid in illustrating how a point cloud
can be very flat in one direction–which is where PCA
comes in to choose a direction that is not flat.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: Gael Varoquaux</span>
<span class="c1"># Jaques Grobler</span>
<span class="c1"># Kevin Hughes</span>
<span class="c1"># License: BSD 3 clause</span>
</pre></div>
</div>
<section id="create-the-data">
<h2>Create the data<a class="headerlink" href="#create-the-data" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default 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">scipy</span> <span class="kn">import</span> <span class="n">stats</span>
<span class="n">e</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.exp.html#numpy.exp" title="numpy.exp" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.norm" title="scipy.stats.norm" class="sphx-glr-backref-module-scipy-stats sphx-glr-backref-type-py-data"><span class="n">stats</span><span class="o">.</span><span class="n">norm</span></a><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.25</span> <span class="o">/</span> <span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.norm" title="scipy.stats.norm" class="sphx-glr-backref-module-scipy-stats sphx-glr-backref-type-py-data"><span class="n">stats</span><span class="o">.</span><span class="n">norm</span></a><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mi">4</span> <span class="o">/</span> <span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">30000</span><span class="p">))</span>
<span class="n">x</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">30000</span><span class="p">))</span>
<span class="n">z</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">density</span> <span class="o">=</span> <span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">pdf</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">pdf_z</span> <span class="o">=</span> <span class="n">pdf</span><span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>
<span class="n">density</span> <span class="o">*=</span> <span class="n">pdf_z</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">y</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">z</span>
<span class="n">norm</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.sqrt.html#numpy.sqrt" title="numpy.sqrt" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span></a><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">var</span><span class="p">()</span> <span class="o">+</span> <span class="n">b</span><span class="o">.</span><span class="n">var</span><span class="p">())</span>
<span class="n">a</span> <span class="o">/=</span> <span class="n">norm</span>
<span class="n">b</span> <span class="o">/=</span> <span class="n">norm</span>
</pre></div>
</div>
</section>
<section id="plot-the-figures">
<h2>Plot the figures<a class="headerlink" href="#plot-the-figures" title="Permalink to this heading">¶</a></h2>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></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="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="c1"># unused but required import for doing 3d projections with matplotlib < 3.2</span>
<span class="kn">import</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="c1"># noqa: F401</span>
<span class="k">def</span> <span class="nf">plot_figs</span><span class="p">(</span><span class="n">fig_num</span><span class="p">,</span> <span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="p">):</span>
<span class="n">fig</span> <span class="o">=</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><span class="n">fig_num</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.clf.html#matplotlib.pyplot.clf" title="matplotlib.pyplot.clf" 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">clf</span></a><span class="p">()</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="s2">"3d"</span><span class="p">,</span> <span class="n">elev</span><span class="o">=</span><span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="o">=</span><span class="n">azim</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_position</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="mf">0.95</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">a</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">b</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">c</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">density</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s2">"+"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span>
<span class="n">Y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.c_.html#numpy.c_" title="numpy.c_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">c_</span></a><span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]</span>
<span class="c1"># Using SciPy's SVD, this would be:</span>
<span class="c1"># _, pca_score, Vt = scipy.linalg.svd(Y, full_matrices=False)</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">n_components</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">pca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="n">V</span> <span class="o">=</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="n">x_pca_axis</span><span class="p">,</span> <span class="n">y_pca_axis</span><span class="p">,</span> <span class="n">z_pca_axis</span> <span class="o">=</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">V</span>
<span class="n">x_pca_plane</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.r_.html#numpy.r_" title="numpy.r_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">r_</span></a><span class="p">[</span><span class="n">x_pca_axis</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span><span class="n">x_pca_axis</span><span class="p">[</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="n">y_pca_plane</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.r_.html#numpy.r_" title="numpy.r_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">r_</span></a><span class="p">[</span><span class="n">y_pca_axis</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span><span class="n">y_pca_axis</span><span class="p">[</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="n">z_pca_plane</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.r_.html#numpy.r_" title="numpy.r_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">r_</span></a><span class="p">[</span><span class="n">z_pca_axis</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span><span class="n">z_pca_axis</span><span class="p">[</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="n">x_pca_plane</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</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">y_pca_plane</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</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">z_pca_plane</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</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">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">x_pca_plane</span><span class="p">,</span> <span class="n">y_pca_plane</span><span class="p">,</span> <span class="n">z_pca_plane</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">zaxis</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([])</span>
<span class="n">elev</span> <span class="o">=</span> <span class="o">-</span><span class="mi">40</span>
<span class="n">azim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">80</span>
<span class="n">plot_figs</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="p">)</span>
<span class="n">elev</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">azim</span> <span class="o">=</span> <span class="mi">20</span>
<span class="n">plot_figs</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" 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">show</span></a><span class="p">()</span>
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