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<li><a class="reference internal" href="#">Ledoit-Wolf vs Covariance simple estimation</a></li>
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<div class="section" id="ledoit-wolf-vs-covariance-simple-estimation">
<span id="example-plot-covariance-estimation-py"></span><h1>Ledoit-Wolf vs Covariance simple estimation<a class="headerlink" href="#ledoit-wolf-vs-covariance-simple-estimation" title="Permalink to this headline">ΒΆ</a></h1>
<p>Covariance estimation can be regularized using a shrinkage parameter.
Ledoit-Wolf estimates automatically this parameter. In this example,
we compute the likelihood of unseen data for different values of
the shrinkage parameter. The Ledoit-Wolf estimate reaches an
almost optimal value.</p>
<img alt="auto_examples/images/plot_covariance_estimation.png" class="align-center" src="auto_examples/images/plot_covariance_estimation.png" />
<p><strong>Python source code:</strong> <a class="reference download internal" href="../_downloads/plot_covariance_estimation.py"><tt class="xref download docutils literal"><span class="pre">plot_covariance_estimation.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="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="c">###############################################################################</span>
<span class="c"># Generate sample data</span>
<span class="n">n_features</span><span class="p">,</span> <span class="n">n_samples</span> <span class="o">=</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">20</span>
<span class="n">X_train</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="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">))</span>
<span class="n">X_test</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="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">))</span>
<span class="c"># Color samples</span>
<span class="n">coloring_matrix</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="n">n_features</span><span class="p">,</span> <span class="n">n_features</span><span class="p">))</span>
<span class="n">X_train</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">X_train</span><span class="p">,</span> <span class="n">coloring_matrix</span><span class="p">)</span>
<span class="n">X_test</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">X_test</span><span class="p">,</span> <span class="n">coloring_matrix</span><span class="p">)</span>
<span class="c">###############################################################################</span>
<span class="c"># Compute Ledoit-Wolf and Covariances on a grid of shrinkages</span>
<span class="kn">from</span> <span class="nn">scikits.learn.covariance</span> <span class="kn">import</span> <span class="n">LedoitWolf</span><span class="p">,</span> <span class="n">ShrunkCovariance</span>
<span class="n">lw</span> <span class="o">=</span> <span class="n">LedoitWolf</span><span class="p">()</span>
<span class="n">loglik_lw</span> <span class="o">=</span> <span class="n">lw</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">ShrunkCovariance</span><span class="p">()</span>
<span class="n">shrinkages</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
<span class="n">negative_logliks</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">cov</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">shrinkage</span><span class="o">=</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span> \
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">shrinkages</span><span class="p">]</span>
<span class="c">###############################################################################</span>
<span class="c"># Plot results</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">loglog</span><span class="p">(</span><span class="n">shrinkages</span><span class="p">,</span> <span class="n">negative_logliks</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">'Shrinkage'</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">'Negative log-likelihood'</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="n">lw</span><span class="o">.</span><span class="n">shrinkage_</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="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">loglik_lw</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">'g'</span><span class="p">,</span>
<span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">'Ledoit-Wolf estimate'</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="n">pl</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
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