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<li><a class="reference internal" href="#">Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood</a><ul>
<li><a class="reference internal" href="#generate-sample-data">Generate sample data</a></li>
<li><a class="reference internal" href="#compute-the-likelihood-on-test-data">Compute the likelihood on test data</a></li>
<li><a class="reference internal" href="#compare-different-approaches-to-setting-the-regularization-parameter">Compare different approaches to setting the regularization parameter</a></li>
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<p><a class="reference internal" href="#sphx-glr-download-auto-examples-covariance-plot-covariance-estimation-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code or to run this example in your browser via JupyterLite or Binder</p>
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<section class="sphx-glr-example-title" id="shrinkage-covariance-estimation-ledoitwolf-vs-oas-and-max-likelihood">
<span id="sphx-glr-auto-examples-covariance-plot-covariance-estimation-py"></span><h1>Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood<a class="headerlink" href="#shrinkage-covariance-estimation-ledoitwolf-vs-oas-and-max-likelihood" title="Link to this heading">¶</a></h1>
<p>When working with covariance estimation, the usual approach is to use
a maximum likelihood estimator, such as the
<a class="reference internal" href="../../modules/generated/sklearn.covariance.EmpiricalCovariance.html#sklearn.covariance.EmpiricalCovariance" title="sklearn.covariance.EmpiricalCovariance"><code class="xref py py-class docutils literal notranslate"><span class="pre">EmpiricalCovariance</span></code></a>. It is unbiased, i.e. it
converges to the true (population) covariance when given many
observations. However, it can also be beneficial to regularize it, in
order to reduce its variance; this, in turn, introduces some bias. This
example illustrates the simple regularization used in
<a class="reference internal" href="../../modules/covariance.html#shrunk-covariance"><span class="std std-ref">Shrunk Covariance</span></a> estimators. In particular, it focuses on how to
set the amount of regularization, i.e. how to choose the bias-variance
trade-off.</p>
<section id="generate-sample-data">
<h2>Generate sample data<a class="headerlink" href="#generate-sample-data" title="Link to this heading">¶</a></h2>
<div class="highlight-Python 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="n">n_features</span><span class="p">,</span> <span class="n">n_samples</span> <span class="o">=</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">20</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">42</span><span class="p">)</span>
<span class="n">base_X_train</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">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">base_X_test</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">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="c1"># Color samples</span>
<span class="n">coloring_matrix</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">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> <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><span class="n">base_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> <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><span class="n">base_X_test</span><span class="p">,</span> <span class="n">coloring_matrix</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="compute-the-likelihood-on-test-data">
<h2>Compute the likelihood on test data<a class="headerlink" href="#compute-the-likelihood-on-test-data" title="Link to this heading">¶</a></h2>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>
<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.covariance.ShrunkCovariance.html#sklearn.covariance.ShrunkCovariance" title="sklearn.covariance.ShrunkCovariance" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ShrunkCovariance</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.covariance.empirical_covariance.html#sklearn.covariance.empirical_covariance" title="sklearn.covariance.empirical_covariance" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-function"><span class="n">empirical_covariance</span></a><span class="p">,</span> <span class="n">log_likelihood</span>
<span class="c1"># spanning a range of possible shrinkage coefficient values</span>
<span class="n">shrinkages</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.logspace.html#numpy.logspace" title="numpy.logspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">logspace</span></a><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><a href="../../modules/generated/sklearn.covariance.ShrunkCovariance.html#sklearn.covariance.ShrunkCovariance" title="sklearn.covariance.ShrunkCovariance" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ShrunkCovariance</span></a><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">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="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">shrinkages</span>
<span class="p">]</span>
<span class="c1"># under the ground-truth model, which we would not have access to in real</span>
<span class="c1"># settings</span>
<span class="n">real_cov</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><span class="n">coloring_matrix</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">coloring_matrix</span><span class="p">)</span>
<span class="n">emp_cov</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.covariance.empirical_covariance.html#sklearn.covariance.empirical_covariance" title="sklearn.covariance.empirical_covariance" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-function"><span class="n">empirical_covariance</span></a><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
<span class="n">loglik_real</span> <span class="o">=</span> <span class="o">-</span><span class="n">log_likelihood</span><span class="p">(</span><span class="n">emp_cov</span><span class="p">,</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.inv.html#scipy.linalg.inv" title="scipy.linalg.inv" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span></a><span class="p">(</span><span class="n">real_cov</span><span class="p">))</span>
</pre></div>
</div>
</section>
<section id="compare-different-approaches-to-setting-the-regularization-parameter">
<h2>Compare different approaches to setting the regularization parameter<a class="headerlink" href="#compare-different-approaches-to-setting-the-regularization-parameter" title="Link to this heading">¶</a></h2>
<p>Here we compare 3 approaches:</p>
<ul class="simple">
<li><p>Setting the parameter by cross-validating the likelihood on three folds
according to a grid of potential shrinkage parameters.</p></li>
<li><p>A close formula proposed by Ledoit and Wolf to compute
the asymptotically optimal regularization parameter (minimizing a MSE
criterion), yielding the <a class="reference internal" href="../../modules/generated/sklearn.covariance.LedoitWolf.html#sklearn.covariance.LedoitWolf" title="sklearn.covariance.LedoitWolf"><code class="xref py py-class docutils literal notranslate"><span class="pre">LedoitWolf</span></code></a>
covariance estimate.</p></li>
<li><p>An improvement of the Ledoit-Wolf shrinkage, the
<a class="reference internal" href="../../modules/generated/sklearn.covariance.OAS.html#sklearn.covariance.OAS" title="sklearn.covariance.OAS"><code class="xref py py-class docutils literal notranslate"><span class="pre">OAS</span></code></a>, proposed by Chen et al. Its
convergence is significantly better under the assumption that the data
are Gaussian, in particular for small samples.</p></li>
</ul>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><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>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSearchCV</span></a>
<span class="c1"># GridSearch for an optimal shrinkage coefficient</span>
<span class="n">tuned_parameters</span> <span class="o">=</span> <span class="p">[{</span><span class="s2">"shrinkage"</span><span class="p">:</span> <span class="n">shrinkages</span><span class="p">}]</span>
<span class="n">cv</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">GridSearchCV</span></a><span class="p">(</span><a href="../../modules/generated/sklearn.covariance.ShrunkCovariance.html#sklearn.covariance.ShrunkCovariance" title="sklearn.covariance.ShrunkCovariance" class="sphx-glr-backref-module-sklearn-covariance sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">ShrunkCovariance</span></a><span class="p">(),</span> <span class="n">tuned_parameters</span><span class="p">)</span>
<span class="n">cv</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="c1"># Ledoit-Wolf optimal shrinkage coefficient estimate</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">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="c1"># OAS coefficient estimate</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">loglik_oa</span> <span class="o">=</span> <span class="n">oa</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>
</pre></div>
</div>
</section>
<section id="plot-results">
<h2>Plot results<a class="headerlink" href="#plot-results" title="Link to this heading">¶</a></h2>
<p>To quantify estimation error, we plot the likelihood of unseen data for
different values of the shrinkage parameter. We also show the choices by
cross-validation, or with the LedoitWolf and OAS estimates.</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="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>
<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">"Regularized covariance: likelihood and shrinkage coefficient"</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">"Regularization parameter: shrinkage coefficient"</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">"Error: negative log-likelihood on test data"</span><span class="p">)</span>
<span class="c1"># range shrinkage curve</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.loglog.html#matplotlib.pyplot.loglog" title="matplotlib.pyplot.loglog" 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">loglog</span></a><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">label</span><span class="o">=</span><span class="s2">"Negative log-likelihood"</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" 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">plot</span></a><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">2</span> <span class="o">*</span> <span class="p">[</span><span class="n">loglik_real</span><span class="p">],</span> <span class="s2">"--r"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Real covariance likelihood"</span><span class="p">)</span>
<span class="c1"># adjust view</span>
<span class="n">lik_max</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.amax.html#numpy.amax" title="numpy.amax" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">amax</span></a><span class="p">(</span><span class="n">negative_logliks</span><span class="p">)</span>
<span class="n">lik_min</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.amin.html#numpy.amin" title="numpy.amin" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">amin</span></a><span class="p">(</span><span class="n">negative_logliks</span><span class="p">)</span>
<span class="n">ymin</span> <span class="o">=</span> <span class="n">lik_min</span> <span class="o">-</span> <span class="mf">6.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.log.html#numpy.log" title="numpy.log" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">log</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">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="n">ymax</span> <span class="o">=</span> <span class="n">lik_max</span> <span class="o">+</span> <span class="mf">10.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.log.html#numpy.log" title="numpy.log" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">log</span></a><span class="p">(</span><span class="n">lik_max</span> <span class="o">-</span> <span class="n">lik_min</span><span class="p">)</span>
<span class="n">xmin</span> <span class="o">=</span> <span class="n">shrinkages</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">xmax</span> <span class="o">=</span> <span class="n">shrinkages</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="c1"># LW likelihood</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.vlines.html#matplotlib.pyplot.vlines" title="matplotlib.pyplot.vlines" 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">vlines</span></a><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">ymin</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="s2">"magenta"</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="s2">"Ledoit-Wolf estimate"</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># OAS likelihood</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.vlines.html#matplotlib.pyplot.vlines" title="matplotlib.pyplot.vlines" 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">vlines</span></a><span class="p">(</span>
<span class="n">oa</span><span class="o">.</span><span class="n">shrinkage_</span><span class="p">,</span> <span class="n">ymin</span><span class="p">,</span> <span class="o">-</span><span class="n">loglik_oa</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"purple"</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="s2">"OAS estimate"</span>
<span class="p">)</span>
<span class="c1"># best CV estimator likelihood</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.vlines.html#matplotlib.pyplot.vlines" title="matplotlib.pyplot.vlines" 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">vlines</span></a><span class="p">(</span>
<span class="n">cv</span><span class="o">.</span><span class="n">best_estimator_</span><span class="o">.</span><span class="n">shrinkage</span><span class="p">,</span>
<span class="n">ymin</span><span class="p">,</span>
<span class="o">-</span><span class="n">cv</span><span class="o">.</span><span class="n">best_estimator_</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">color</span><span class="o">=</span><span class="s2">"cyan"</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="s2">"Cross-validation best estimate"</span><span class="p">,</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="n">ymin</span><span class="p">,</span> <span class="n">ymax</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="n">xmin</span><span class="p">,</span> <span class="n">xmax</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>
<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>
</div>
<img src="../../_images/sphx_glr_plot_covariance_estimation_001.png" srcset="../../_images/sphx_glr_plot_covariance_estimation_001.png" alt="Regularized covariance: likelihood and shrinkage coefficient" class = "sphx-glr-single-img"/><div class="admonition note">
<p class="admonition-title">Note</p>
<p>The maximum likelihood estimate corresponds to no shrinkage,
and thus performs poorly. The Ledoit-Wolf estimate performs really well,
as it is close to the optimal and is not computationally costly. In this
example, the OAS estimate is a bit further away. Interestingly, both
approaches outperform cross-validation, which is significantly most
computationally costly.</p>
</div>
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<p class="rubric">Related examples</p>
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