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<li><a class="reference internal" href="#">Ledoit-Wolf vs OAS estimation</a></li>
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<div class="sphx-glr-download-link-note admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Click <a class="reference internal" href="#sphx-glr-download-auto-examples-covariance-plot-lw-vs-oas-py"><span class="std std-ref">here</span></a> to download the full example code</p>
</div>
<div class="sphx-glr-example-title section" id="ledoit-wolf-vs-oas-estimation">
<span id="sphx-glr-auto-examples-covariance-plot-lw-vs-oas-py"></span><h1>Ledoit-Wolf vs OAS estimation<a class="headerlink" href="#ledoit-wolf-vs-oas-estimation" title="Permalink to this headline">¶</a></h1>
<p>The usual covariance maximum likelihood estimate can be regularized
using shrinkage. Ledoit and Wolf proposed a close formula to compute
the asymptotically optimal shrinkage parameter (minimizing a MSE
criterion), yielding the Ledoit-Wolf covariance estimate.</p>
<p>Chen et al. proposed an improvement of the Ledoit-Wolf shrinkage
parameter, the OAS coefficient, whose convergence is significantly
better under the assumption that the data are Gaussian.</p>
<p>This example, inspired from Chen’s publication [1], shows a comparison
of the estimated MSE of the LW and OAS methods, using Gaussian
distributed data.</p>
<p>[1] “Shrinkage Algorithms for MMSE Covariance Estimation”
Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">scipy.linalg</span> <span class="k">import</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.toeplitz.html#scipy.linalg.toeplitz" title="View documentation for scipy.linalg.toeplitz"><span class="n">toeplitz</span></a><span class="p">,</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.cholesky.html#scipy.linalg.cholesky" title="View documentation for scipy.linalg.cholesky"><span class="n">cholesky</span></a>
<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="k">import</span> <a href="../../modules/generated/sklearn.covariance.LedoitWolf.html#sklearn.covariance.LedoitWolf" title="View documentation for sklearn.covariance.LedoitWolf"><span class="n">LedoitWolf</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.covariance.OAS.html#sklearn.covariance.OAS" title="View documentation for sklearn.covariance.OAS"><span class="n">OAS</span></a>
<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">0</span><span class="p">)</span>
</pre></div>
</div>
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">n_features</span> <span class="o">=</span> <span class="mi">100</span>
<span class="c1"># simulation covariance matrix (AR(1) process)</span>
<span class="n">r</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">real_cov</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.toeplitz.html#scipy.linalg.toeplitz" title="View documentation for scipy.linalg.toeplitz"><span class="n">toeplitz</span></a><span class="p">(</span><span class="n">r</span> <span class="o">**</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.arange.html#numpy.arange" title="View documentation for numpy.arange"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="n">n_features</span><span class="p">))</span>
<span class="n">coloring_matrix</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.cholesky.html#scipy.linalg.cholesky" title="View documentation for scipy.linalg.cholesky"><span class="n">cholesky</span></a><span class="p">(</span><span class="n">real_cov</span><span class="p">)</span>
<span class="n">n_samples_range</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.arange.html#numpy.arange" title="View documentation for numpy.arange"><span class="n">np</span><span class="o">.</span><span class="n">arange</span></a><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">repeat</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">lw_mse</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_samples_range</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">repeat</span><span class="p">))</span>
<span class="n">oa_mse</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_samples_range</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">repeat</span><span class="p">))</span>
<span class="n">lw_shrinkage</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_samples_range</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">repeat</span><span class="p">))</span>
<span class="n">oa_shrinkage</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_samples_range</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">repeat</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n_samples</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">n_samples_range</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">repeat</span><span class="p">):</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html#numpy.dot" title="View documentation for numpy.dot"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</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">coloring_matrix</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="n">lw</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.covariance.LedoitWolf.html#sklearn.covariance.LedoitWolf" title="View documentation for sklearn.covariance.LedoitWolf"><span class="n">LedoitWolf</span></a><span class="p">(</span><span class="n">store_precision</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">assume_centered</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">lw</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="n">lw_mse</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">lw</span><span class="o">.</span><span class="n">error_norm</span><span class="p">(</span><span class="n">real_cov</span><span class="p">,</span> <span class="n">scaling</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">lw_shrinkage</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">lw</span><span class="o">.</span><span class="n">shrinkage_</span>
<span class="n">oa</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.covariance.OAS.html#sklearn.covariance.OAS" title="View documentation for sklearn.covariance.OAS"><span class="n">OAS</span></a><span class="p">(</span><span class="n">store_precision</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">assume_centered</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">oa</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="n">oa_mse</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">oa</span><span class="o">.</span><span class="n">error_norm</span><span class="p">(</span><span class="n">real_cov</span><span class="p">,</span> <span class="n">scaling</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">oa_shrinkage</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">oa</span><span class="o">.</span><span class="n">shrinkage_</span>
<span class="c1"># plot MSE</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><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">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="View documentation for matplotlib.pyplot.errorbar"><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span></a><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">lw_mse</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">yerr</span><span class="o">=</span><span class="n">lw_mse</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'Ledoit-Wolf'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'navy'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="View documentation for matplotlib.pyplot.errorbar"><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span></a><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">oa_mse</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">yerr</span><span class="o">=</span><span class="n">oa_mse</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'OAS'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'darkorange'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Squared error"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><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">"upper right"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s2">"Comparison of covariance estimators"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="View documentation for matplotlib.pyplot.xlim"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span></a><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">31</span><span class="p">)</span>
<span class="c1"># plot shrinkage coefficient</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="View documentation for matplotlib.pyplot.errorbar"><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span></a><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">lw_shrinkage</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">yerr</span><span class="o">=</span><span class="n">lw_shrinkage</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'Ledoit-Wolf'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'navy'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="View documentation for matplotlib.pyplot.errorbar"><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span></a><span class="p">(</span><span class="n">n_samples_range</span><span class="p">,</span> <span class="n">oa_shrinkage</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">yerr</span><span class="o">=</span><span class="n">oa_shrinkage</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span>
<span class="n">label</span><span class="o">=</span><span class="s1">'OAS'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'darkorange'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s2">"n_samples"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s2">"Shrinkage"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="View documentation for matplotlib.pyplot.legend"><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>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="View documentation for matplotlib.pyplot.ylim"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">(</span><a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="View documentation for matplotlib.pyplot.ylim"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">1.</span> <span class="o">+</span> <span class="p">(</span><a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="View documentation for matplotlib.pyplot.ylim"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">()[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="View documentation for matplotlib.pyplot.ylim"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">()[</span><span class="mi">0</span><span class="p">])</span> <span class="o">/</span> <span class="mf">10.</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="View documentation for matplotlib.pyplot.xlim"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span></a><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">31</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
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