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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.covariance</span></code>.OAS</a><ul>
<li><a class="reference internal" href="#examples-using-sklearn-covariance-oas">Examples using <code class="docutils literal notranslate"><span class="pre">sklearn.covariance.OAS</span></code></a></li>
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<div class="section" id="sklearn-covariance-oas">
<h1><a class="reference internal" href="../classes.html#module-sklearn.covariance" title="sklearn.covariance"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.covariance</span></code></a>.OAS<a class="headerlink" href="#sklearn-covariance-oas" title="Permalink to this headline">¶</a></h1>
<dl class="py class">
<dt id="sklearn.covariance.OAS">
<em class="property">class </em><code class="sig-prename descclassname">sklearn.covariance.</code><code class="sig-name descname">OAS</code><span class="sig-paren">(</span><em class="sig-param"><span class="o">*</span></em>, <em class="sig-param"><span class="n">store_precision</span><span class="o">=</span><span class="default_value">True</span></em>, <em class="sig-param"><span class="n">assume_centered</span><span class="o">=</span><span class="default_value">False</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/covariance/_shrunk_covariance.py#L499"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS" title="Permalink to this definition">¶</a></dt>
<dd><p>Oracle Approximating Shrinkage Estimator</p>
<p>Read more in the <a class="reference internal" href="../covariance.html#shrunk-covariance"><span class="std std-ref">User Guide</span></a>.</p>
<p>OAS is a particular form of shrinkage described in
“Shrinkage Algorithms for MMSE Covariance Estimation”
Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.</p>
<p>The formula used here does not correspond to the one given in the
article. In the original article, formula (23) states that 2/p is
multiplied by Trace(cov*cov) in both the numerator and denominator, but
this operation is omitted because for a large p, the value of 2/p is
so small that it doesn’t affect the value of the estimator.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>store_precision</strong><span class="classifier">bool, default=True</span></dt><dd><p>Specify if the estimated precision is stored.</p>
</dd>
<dt><strong>assume_centered</strong><span class="classifier">bool, default=False</span></dt><dd><p>If True, data will not be centered before computation.
Useful when working with data whose mean is almost, but not exactly
zero.
If False (default), data will be centered before computation.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Attributes</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>covariance_</strong><span class="classifier">ndarray of shape (n_features, n_features)</span></dt><dd><p>Estimated covariance matrix.</p>
</dd>
<dt><strong>location_</strong><span class="classifier">ndarray of shape (n_features,)</span></dt><dd><p>Estimated location, i.e. the estimated mean.</p>
</dd>
<dt><strong>precision_</strong><span class="classifier">ndarray of shape (n_features, n_features)</span></dt><dd><p>Estimated pseudo inverse matrix.
(stored only if store_precision is True)</p>
</dd>
<dt><strong>shrinkage_</strong><span class="classifier">float</span></dt><dd><p>coefficient in the convex combination used for the computation
of the shrunk estimate. Range is [0, 1].</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The regularised covariance is:</p>
<p>(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)</p>
<p>where mu = trace(cov) / n_features
and shrinkage is given by the OAS formula (see References)</p>
<p class="rubric">References</p>
<p>“Shrinkage Algorithms for MMSE Covariance Estimation”
Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <span class="n">OAS</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_gaussian_quantiles</span>
<span class="gp">>>> </span><span class="n">real_cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">.</span><span class="mi">8</span><span class="p">,</span> <span class="o">.</span><span class="mi">3</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="o">.</span><span class="mi">3</span><span class="p">,</span> <span class="o">.</span><span class="mi">4</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">rng</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">RandomState</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="o">=</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="gp">... </span> <span class="n">cov</span><span class="o">=</span><span class="n">real_cov</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">size</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">oas</span> <span class="o">=</span> <span class="n">OAS</span><span class="p">()</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="gp">>>> </span><span class="n">oas</span><span class="o">.</span><span class="n">covariance_</span>
<span class="go">array([[0.7533..., 0.2763...],</span>
<span class="go"> [0.2763..., 0.3964...]])</span>
<span class="gp">>>> </span><span class="n">oas</span><span class="o">.</span><span class="n">precision_</span>
<span class="go">array([[ 1.7833..., -1.2431... ],</span>
<span class="go"> [-1.2431..., 3.3889...]])</span>
<span class="gp">>>> </span><span class="n">oas</span><span class="o">.</span><span class="n">shrinkage_</span>
<span class="go">0.0195...</span>
</pre></div>
</div>
<p class="rubric">Methods</p>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.error_norm" title="sklearn.covariance.OAS.error_norm"><code class="xref py py-obj docutils literal notranslate"><span class="pre">error_norm</span></code></a>(comp_cov[, norm, scaling, squared])</p></td>
<td><p>Computes the Mean Squared Error between two covariance estimators.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.fit" title="sklearn.covariance.OAS.fit"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fit</span></code></a>(X[, y])</p></td>
<td><p>Fit the Oracle Approximating Shrinkage covariance model according to the given training data and parameters.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.get_params" title="sklearn.covariance.OAS.get_params"><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_params</span></code></a>([deep])</p></td>
<td><p>Get parameters for this estimator.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.get_precision" title="sklearn.covariance.OAS.get_precision"><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_precision</span></code></a>()</p></td>
<td><p>Getter for the precision matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.mahalanobis" title="sklearn.covariance.OAS.mahalanobis"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mahalanobis</span></code></a>(X)</p></td>
<td><p>Computes the squared Mahalanobis distances of given observations.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.score" title="sklearn.covariance.OAS.score"><code class="xref py py-obj docutils literal notranslate"><span class="pre">score</span></code></a>(X_test[, y])</p></td>
<td><p>Computes the log-likelihood of a Gaussian data set with <code class="docutils literal notranslate"><span class="pre">self.covariance_</span></code> as an estimator of its covariance matrix.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#sklearn.covariance.OAS.set_params" title="sklearn.covariance.OAS.set_params"><code class="xref py py-obj docutils literal notranslate"><span class="pre">set_params</span></code></a>(**params)</p></td>
<td><p>Set the parameters of this estimator.</p></td>
</tr>
</tbody>
</table>
<dl class="py method">
<dt id="sklearn.covariance.OAS.error_norm">
<code class="sig-name descname">error_norm</code><span class="sig-paren">(</span><em class="sig-param"><span class="n">comp_cov</span></em>, <em class="sig-param"><span class="n">norm</span><span class="o">=</span><span class="default_value">'frobenius'</span></em>, <em class="sig-param"><span class="n">scaling</span><span class="o">=</span><span class="default_value">True</span></em>, <em class="sig-param"><span class="n">squared</span><span class="o">=</span><span class="default_value">True</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/covariance/_empirical_covariance.py#L245"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.error_norm" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the Mean Squared Error between two covariance estimators.
(In the sense of the Frobenius norm).</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>comp_cov</strong><span class="classifier">array-like of shape (n_features, n_features)</span></dt><dd><p>The covariance to compare with.</p>
</dd>
<dt><strong>norm</strong><span class="classifier">{“frobenius”, “spectral”}, default=”frobenius”</span></dt><dd><p>The type of norm used to compute the error. Available error types:
- ‘frobenius’ (default): sqrt(tr(A^t.A))
- ‘spectral’: sqrt(max(eigenvalues(A^t.A))
where A is the error <code class="docutils literal notranslate"><span class="pre">(comp_cov</span> <span class="pre">-</span> <span class="pre">self.covariance_)</span></code>.</p>
</dd>
<dt><strong>scaling</strong><span class="classifier">bool, default=True</span></dt><dd><p>If True (default), the squared error norm is divided by n_features.
If False, the squared error norm is not rescaled.</p>
</dd>
<dt><strong>squared</strong><span class="classifier">bool, default=True</span></dt><dd><p>Whether to compute the squared error norm or the error norm.
If True (default), the squared error norm is returned.
If False, the error norm is returned.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>result</strong><span class="classifier">float</span></dt><dd><p>The Mean Squared Error (in the sense of the Frobenius norm) between
<code class="docutils literal notranslate"><span class="pre">self</span></code> and <code class="docutils literal notranslate"><span class="pre">comp_cov</span></code> covariance estimators.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt id="sklearn.covariance.OAS.fit">
<code class="sig-name descname">fit</code><span class="sig-paren">(</span><em class="sig-param"><span class="n">X</span></em>, <em class="sig-param"><span class="n">y</span><span class="o">=</span><span class="default_value">None</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/covariance/_shrunk_covariance.py#L577"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.fit" title="Permalink to this definition">¶</a></dt>
<dd><p>Fit the Oracle Approximating Shrinkage covariance model
according to the given training data and parameters.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like of shape (n_samples, n_features)</span></dt><dd><p>Training data, where <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> is the number of samples
and <code class="docutils literal notranslate"><span class="pre">n_features</span></code> is the number of features.</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Not used, present for API consistency by convention.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>self</strong><span class="classifier">object</span></dt><dd></dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt id="sklearn.covariance.OAS.get_params">
<code class="sig-name descname">get_params</code><span class="sig-paren">(</span><em class="sig-param"><span class="n">deep</span><span class="o">=</span><span class="default_value">True</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/base.py#L178"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.get_params" title="Permalink to this definition">¶</a></dt>
<dd><p>Get parameters for this estimator.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>deep</strong><span class="classifier">bool, default=True</span></dt><dd><p>If True, will return the parameters for this estimator and
contained subobjects that are estimators.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>params</strong><span class="classifier">dict</span></dt><dd><p>Parameter names mapped to their values.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt id="sklearn.covariance.OAS.get_precision">
<code class="sig-name descname">get_precision</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/covariance/_empirical_covariance.py#L174"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.get_precision" title="Permalink to this definition">¶</a></dt>
<dd><p>Getter for the precision matrix.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>precision_</strong><span class="classifier">array-like of shape (n_features, n_features)</span></dt><dd><p>The precision matrix associated to the current covariance object.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt id="sklearn.covariance.OAS.mahalanobis">
<code class="sig-name descname">mahalanobis</code><span class="sig-paren">(</span><em class="sig-param"><span class="n">X</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/covariance/_empirical_covariance.py#L297"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.mahalanobis" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the squared Mahalanobis distances of given observations.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X</strong><span class="classifier">array-like of shape (n_samples, n_features)</span></dt><dd><p>The observations, the Mahalanobis distances of the which we
compute. Observations are assumed to be drawn from the same
distribution than the data used in fit.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>dist</strong><span class="classifier">ndarray of shape (n_samples,)</span></dt><dd><p>Squared Mahalanobis distances of the observations.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt id="sklearn.covariance.OAS.score">
<code class="sig-name descname">score</code><span class="sig-paren">(</span><em class="sig-param"><span class="n">X_test</span></em>, <em class="sig-param"><span class="n">y</span><span class="o">=</span><span class="default_value">None</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/covariance/_empirical_covariance.py#L216"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.score" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the log-likelihood of a Gaussian data set with
<code class="docutils literal notranslate"><span class="pre">self.covariance_</span></code> as an estimator of its covariance matrix.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>X_test</strong><span class="classifier">array-like of shape (n_samples, n_features)</span></dt><dd><p>Test data of which we compute the likelihood, where n_samples is
the number of samples and n_features is the number of features.
X_test is assumed to be drawn from the same distribution than
the data used in fit (including centering).</p>
</dd>
<dt><strong>y</strong><span class="classifier">Ignored</span></dt><dd><p>Not used, present for API consistency by convention.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>res</strong><span class="classifier">float</span></dt><dd><p>The likelihood of the data set with <code class="docutils literal notranslate"><span class="pre">self.covariance_</span></code> as an
estimator of its covariance matrix.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt id="sklearn.covariance.OAS.set_params">
<code class="sig-name descname">set_params</code><span class="sig-paren">(</span><em class="sig-param"><span class="o">**</span><span class="n">params</span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/23d876161/sklearn/base.py#L202"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sklearn.covariance.OAS.set_params" title="Permalink to this definition">¶</a></dt>
<dd><p>Set the parameters of this estimator.</p>
<p>The method works on simple estimators as well as on nested objects
(such as <a class="reference internal" href="sklearn.pipeline.Pipeline.html#sklearn.pipeline.Pipeline" title="sklearn.pipeline.Pipeline"><code class="xref py py-class docutils literal notranslate"><span class="pre">Pipeline</span></code></a>). The latter have
parameters of the form <code class="docutils literal notranslate"><span class="pre"><component>__<parameter></span></code> so that it’s
possible to update each component of a nested object.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>**params</strong><span class="classifier">dict</span></dt><dd><p>Estimator parameters.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>self</strong><span class="classifier">estimator instance</span></dt><dd><p>Estimator instance.</p>
</dd>
</dl>
</dd>
</dl>
</dd></dl>
</dd></dl>
<div class="section" id="examples-using-sklearn-covariance-oas">
<h2>Examples using <code class="docutils literal notranslate"><span class="pre">sklearn.covariance.OAS</span></code><a class="headerlink" href="#examples-using-sklearn-covariance-oas" title="Permalink to this headline">¶</a></h2>
<div class="sphx-glr-thumbcontainer" tooltip="This example illustrates how the Ledoit-Wolf and Oracle Shrinkage Approximating (OAS) estimator..."><div class="figure align-default" id="id1">
<img alt="Normal, Ledoit-Wolf and OAS Linear Discriminant Analysis for classification" src="../../_images/sphx_glr_plot_lda_thumb.png" />
<p class="caption"><span class="caption-text"><a class="reference internal" href="../../auto_examples/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></span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="The usual covariance maximum likelihood estimate can be regularized using shrinkage. Ledoit and..."><div class="figure align-default" id="id2">
<img alt="Ledoit-Wolf vs OAS estimation" src="../../_images/sphx_glr_plot_lw_vs_oas_thumb.png" />
<p class="caption"><span class="caption-text"><a class="reference internal" href="../../auto_examples/covariance/plot_lw_vs_oas.html#sphx-glr-auto-examples-covariance-plot-lw-vs-oas-py"><span class="std std-ref">Ledoit-Wolf vs OAS estimation</span></a></span><a class="headerlink" href="#id2" title="Permalink to this image">¶</a></p>
</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="When working with covariance estimation, the usual approach is to use a maximum likelihood esti..."><div class="figure align-default" id="id3">
<img alt="Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood" src="../../_images/sphx_glr_plot_covariance_estimation_thumb.png" />
<p class="caption"><span class="caption-text"><a class="reference internal" href="../../auto_examples/covariance/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></span><a class="headerlink" href="#id3" title="Permalink to this image">¶</a></p>
</div>
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