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<li><a class="reference internal" href="#">Ledoit-Wolf vs OAS estimation</a></li>
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<section class="sphx-glr-example-title" 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="Link to this heading">¶</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-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="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.linalg</span> <span class="kn">import</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.cholesky.html#scipy.linalg.cholesky" title="scipy.linalg.cholesky" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><span class="n">cholesky</span></a><span class="p">,</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.toeplitz.html#scipy.linalg.toeplitz" title="scipy.linalg.toeplitz" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><span class="n">toeplitz</span></a>
<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.covariance.OAS.html#sklearn.covariance.OAS" title="sklearn.covariance.OAS" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">OAS</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.covariance.LedoitWolf.html#sklearn.covariance.LedoitWolf" title="sklearn.covariance.LedoitWolf" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LedoitWolf</span></a>
<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">0</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-Python notranslate"><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="scipy.linalg.toeplitz" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><span class="n">toeplitz</span></a><span class="p">(</span><span class="n">r</span> <span class="o">**</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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="scipy.linalg.cholesky" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><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://numpy.org/doc/stable/reference/generated/numpy.arange.html#numpy.arange" title="numpy.arange" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><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://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><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">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="sklearn.covariance.LedoitWolf" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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="sklearn.covariance.OAS" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><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/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">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="matplotlib.pyplot.errorbar" 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">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="s2">"Ledoit-Wolf"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"navy"</span><span class="p">,</span>
<span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="matplotlib.pyplot.errorbar" 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">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="s2">"OAS"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"darkorange"</span><span class="p">,</span>
<span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</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">"Squared error"</span><span class="p">)</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">"upper right"</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">"Comparison of covariance estimators"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" 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">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/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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="matplotlib.pyplot.errorbar" 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">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="s2">"Ledoit-Wolf"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"navy"</span><span class="p">,</span>
<span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.errorbar.html#matplotlib.pyplot.errorbar" title="matplotlib.pyplot.errorbar" 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">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="s2">"OAS"</span><span class="p">,</span>
<span class="n">color</span><span class="o">=</span><span class="s2">"darkorange"</span><span class="p">,</span>
<span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="p">)</span>
<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">"n_samples"</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">"Shrinkage"</span><span class="p">)</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>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" 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">ylim</span></a><span class="p">(</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" 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">ylim</span></a><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">1.0</span> <span class="o">+</span> <span class="p">(</span><a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" 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">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/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" 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">ylim</span></a><span class="p">()[</span><span class="mi">0</span><span class="p">])</span> <span class="o">/</span> <span class="mf">10.0</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" 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">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/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>
</pre></div>
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<p class="rubric">Related examples</p>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="When working with covariance estimation, the usual approach is to use a maximum likelihood esti..."><img alt="" src="../../_images/sphx_glr_plot_covariance_estimation_thumb.png" />
<p><a class="reference internal" href="plot_covariance_estimation.html#sphx-glr-auto-examples-covariance-plot-covariance-estimation-py"><span class="std std-ref">Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood</span></a></p>
<div class="sphx-glr-thumbnail-title">Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates how the Ledoit-Wolf and Oracle Approximating Shrinkage (OAS) estimator..."><img alt="" src="../../_images/sphx_glr_plot_lda_thumb.png" />
<p><a class="reference internal" href="../classification/plot_lda.html#sphx-glr-auto-examples-classification-plot-lda-py"><span class="std std-ref">Normal, Ledoit-Wolf and OAS Linear Discriminant Analysis for classification</span></a></p>
<div class="sphx-glr-thumbnail-title">Normal, Ledoit-Wolf and OAS Linear Discriminant Analysis for classification</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The usual covariance maximum likelihood estimate is very sensitive to the presence of outliers ..."><img alt="" src="../../_images/sphx_glr_plot_robust_vs_empirical_covariance_thumb.png" />
<p><a class="reference internal" href="plot_robust_vs_empirical_covariance.html#sphx-glr-auto-examples-covariance-plot-robust-vs-empirical-covariance-py"><span class="std std-ref">Robust vs Empirical covariance estimate</span></a></p>
<div class="sphx-glr-thumbnail-title">Robust vs Empirical covariance estimate</div>
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<p><a class="reference internal" href="plot_sparse_cov.html#sphx-glr-auto-examples-covariance-plot-sparse-cov-py"><span class="std std-ref">Sparse inverse covariance estimation</span></a></p>
<div class="sphx-glr-thumbnail-title">Sparse inverse covariance estimation</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="A plot that compares the various convex loss functions supported by SGDClassifier ."><img alt="" src="../../_images/sphx_glr_plot_sgd_loss_functions_thumb.png" />
<p><a class="reference internal" href="../linear_model/plot_sgd_loss_functions.html#sphx-glr-auto-examples-linear-model-plot-sgd-loss-functions-py"><span class="std std-ref">SGD: convex loss functions</span></a></p>
<div class="sphx-glr-thumbnail-title">SGD: convex loss functions</div>
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