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<ul>
<li><a class="reference internal" href="#">2.7. Novelty and Outlier Detection</a><ul>
<li><a class="reference internal" href="#novelty-detection">2.7.1. Novelty Detection</a></li>
<li><a class="reference internal" href="#id1">2.7.2. Outlier Detection</a><ul>
<li><a class="reference internal" href="#fitting-an-elliptic-envelope">2.7.2.1. Fitting an elliptic envelope</a></li>
<li><a class="reference internal" href="#one-class-svm-versus-elliptic-envelope">2.7.2.2. One-class SVM versus elliptic envelope</a></li>
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<div class="section" id="novelty-and-outlier-detection">
<span id="outlier-detection"></span><h1>2.7. Novelty and Outlier Detection<a class="headerlink" href="#novelty-and-outlier-detection" title="Permalink to this headline">¶</a></h1>
<p>Many applications require being able to decide whether a new observation
belongs to the same distribution as existing observations (it is an
<cite>inlier</cite>), or should be considered as different (it is an outlier).
Often, this ability is used to clean real data sets. Two important
distinction must be made:</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name" colspan="2">novelty detection:</th></tr>
<tr class="field-odd field"><td> </td><td class="field-body">The training data is not polluted by outliers, and we are interested in
detecting anomalies in new observations.</td>
</tr>
<tr class="field-even field"><th class="field-name" colspan="2">outlier detection:</th></tr>
<tr class="field-even field"><td> </td><td class="field-body">The training data contains outliers, and we need to fit the central
mode of the training data, ignoring the deviant observations.</td>
</tr>
</tbody>
</table>
<p>The scikit-learn project provides a set of machine learning tools that
can be used both for novelty or outliers detection. This strategy is
implemented with objects learning in an unsupervised way from the data:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
</pre></div>
</div>
<p>new observations can then be sorted as inliers or outliers with a
<cite>predict</cite> method:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
</pre></div>
</div>
<p>Inliers are labeled 1, while outliers are labeled -1.</p>
<div class="section" id="novelty-detection">
<h2>2.7.1. Novelty Detection<a class="headerlink" href="#novelty-detection" title="Permalink to this headline">¶</a></h2>
<p>Consider a data set of <img class="math" src="../_images/math/413f8a8e40062a9090d9d50b88bc7b551b314c26.png" alt="n"/> observations from the same
distribution described by <img class="math" src="../_images/math/3eca8557203e86160952e1c0f735f7417f3285b1.png" alt="p"/> features. Consider now that we
add one more observation to that data set. Is the new observation so
different from the others that we can doubt it is regular? (i.e. does
it come from the same distribution?) Or on the contrary, is it so
similar to the other that we cannot distinguish it from the original
observations? This is the question addressed by the novelty detection
tools and methods.</p>
<p>In general, it is about to learn a rough, close frontier delimiting
the contour of the initial observations distribution, plotted in
embedding <img class="math" src="../_images/math/3eca8557203e86160952e1c0f735f7417f3285b1.png" alt="p"/>-dimensional space. Then, if further observations
lay within the frontier-delimited subspace, they are considered as
coming from the same population than the initial
observations. Otherwise, if they lay outside the frontier, we can say
that they are abnormal with a given confidence in our assessment.</p>
<p>The One-Class SVM has been introduced by Schölkopf et al. for that purpose
and implemented in the <a class="reference internal" href="svm.html#svm"><em>Support Vector Machines</em></a> module in the
<a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a> object. It requires the choice of a
kernel and a scalar parameter to define a frontier. The RBF kernel is
usually chosen although there exists no exact formula or algorithm to
set its bandwidth parameter. This is the default in the scikit-learn
implementation. The <img class="math" src="../_images/math/d67fb61cfffca2cc069e083d76cb6220b4ca14c8.png" alt="\nu"/> parameter, also known as the margin of
the One-Class SVM, corresponds to the probability of finding a new,
but regular, observation outside the frontier.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><a class="reference external" href="http://dl.acm.org/citation.cfm?id=1119749">Estimating the support of a high-dimensional distribution</a> Schölkopf,
Bernhard, et al. Neural computation 13.7 (2001): 1443-1471.</li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li>See <a class="reference internal" href="../auto_examples/svm/plot_oneclass.html#example-svm-plot-oneclass-py"><em>One-class SVM with non-linear kernel (RBF)</em></a> for visualizing the
frontier learned around some data by a
<a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a> object.</li>
</ul>
</div>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/svm/plot_oneclasse.html"><img alt="../_images/plot_oneclass_0011.png" src="../_images/plot_oneclass_0011.png" style="width: 600.0px; height: 450.0px;" /></a>
</div>
</div>
<div class="section" id="id1">
<h2>2.7.2. Outlier Detection<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
<p>Outlier detection is similar to novelty detection in the sense that
the goal is to separate a core of regular observations from some
polluting ones, called “outliers”. Yet, in the case of outlier
detection, we don’t have a clean data set representing the population
of regular observations that can be used to train any tool.</p>
<div class="section" id="fitting-an-elliptic-envelope">
<h3>2.7.2.1. Fitting an elliptic envelope<a class="headerlink" href="#fitting-an-elliptic-envelope" title="Permalink to this headline">¶</a></h3>
<p>One common way of performing outlier detection is to assume that the
regular data come from a known distribution (e.g. data are Gaussian
distributed). From this assumption, we generally try to define the
“shape” of the data, and can define outlying observations as
observations which stand far enough from the fit shape.</p>
<p>The scikit-learn provides an object
<a class="reference internal" href="generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope"><tt class="xref py py-class docutils literal"><span class="pre">covariance.EllipticEnvelope</span></tt></a> that fits a robust covariance
estimate to the data, and thus fits an ellipse to the central data
points, ignoring points outside the central mode.</p>
<p>For instance, assuming that the inlier data are Gaussian distributed, it
will estimate the inlier location and covariance in a robust way (i.e.
whithout being influenced by outliers). The Mahalanobis distances
obtained from this estimate is used to derive a measure of outlyingness.
This strategy is illustrated below.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/covariance/plot_mahalanobis_distances.html"><img alt="../_images/plot_mahalanobis_distances_0011.png" src="../_images/plot_mahalanobis_distances_0011.png" style="width: 600.0px; height: 450.0px;" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li>See <a class="reference internal" href="../auto_examples/covariance/plot_mahalanobis_distances.html#example-covariance-plot-mahalanobis-distances-py"><em>Robust covariance estimation and Mahalanobis distances relevance</em></a> for
an illustration of the difference between using a standard
(<a class="reference internal" href="generated/sklearn.covariance.EmpiricalCovariance.html#sklearn.covariance.EmpiricalCovariance" title="sklearn.covariance.EmpiricalCovariance"><tt class="xref py py-class docutils literal"><span class="pre">covariance.EmpiricalCovariance</span></tt></a>) or a robust estimate
(<a class="reference internal" href="generated/sklearn.covariance.MinCovDet.html#sklearn.covariance.MinCovDet" title="sklearn.covariance.MinCovDet"><tt class="xref py py-class docutils literal"><span class="pre">covariance.MinCovDet</span></tt></a>) of location and covariance to
assess the degree of outlyingness of an observation.</li>
</ul>
</div>
<div class="topic">
<p class="topic-title first">References:</p>
<table class="docutils citation" frame="void" id="rd1999" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label">[RD1999]</td><td>Rousseeuw, P.J., Van Driessen, K. “A fast algorithm for the minimum
covariance determinant estimator” Technometrics 41(3), 212 (1999)</td></tr>
</tbody>
</table>
</div>
</div>
<div class="section" id="one-class-svm-versus-elliptic-envelope">
<h3>2.7.2.2. One-class SVM versus elliptic envelope<a class="headerlink" href="#one-class-svm-versus-elliptic-envelope" title="Permalink to this headline">¶</a></h3>
<p>Strictly-speaking, the One-class SVM is not an outlier-detection method,
but a novelty-detection method: its training set should not be
contaminated by outliers as it may fit them. That said, outlier detection
in high-dimension, or without any assumptions on the distribution of the
inlying data is very challenging, and a One-class SVM gives useful
results in these situations.</p>
<p>The examples below illustrate how the performance of the
<a class="reference internal" href="generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope"><tt class="xref py py-class docutils literal"><span class="pre">covariance.EllipticEnvelope</span></tt></a> degrades as the data is less and
less unimodal. <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a> works better on data with
multiple modes.</p>
<table border="1" class="docutils">
<caption><strong>Comparing One-class SVM approach, and elliptic envelope</strong></caption>
<colgroup>
<col width="40%" />
<col width="60%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td>For a inlier mode well-centered and elliptic, the
<a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a> is not able to benefit from the
rotational symmetry of the inlier population. In addition, it
fits a bit the outliers present in the training set. On the
opposite, the decision rule based on fitting an
<a class="reference internal" href="generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope"><tt class="xref py py-class docutils literal"><span class="pre">covariance.EllipticEnvelope</span></tt></a> learns an ellipse, which
fits well the inlier distribution.</td>
<td><a class="reference external" href="../auto_examples/covariance/plot_outlier_detection.html"><img alt="outlier1" src="../_images/plot_outlier_detection_001.png" style="width: 500.0px; height: 250.0px;" /></a></td>
</tr>
<tr class="row-even"><td>As the inlier distribution becomes bimodal, the
<a class="reference internal" href="generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope"><tt class="xref py py-class docutils literal"><span class="pre">covariance.EllipticEnvelope</span></tt></a> does not fit well the
inliers. However, we can see that the <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a>
tends to overfit: because it has not model of inliers, it
interprets a region where, by chance some outliers are
clustered, as inliers.</td>
<td><a class="reference external" href="../auto_examples/covariance/plot_outlier_detection.html"><img alt="outlier2" src="../_images/plot_outlier_detection_002.png" style="width: 500.0px; height: 250.0px;" /></a></td>
</tr>
<tr class="row-odd"><td>If the inlier distribution is strongly non Gaussian, the
<a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a> is able to recover a reasonable
approximation, whereas the <a class="reference internal" href="generated/sklearn.covariance.EllipticEnvelope.html#sklearn.covariance.EllipticEnvelope" title="sklearn.covariance.EllipticEnvelope"><tt class="xref py py-class docutils literal"><span class="pre">covariance.EllipticEnvelope</span></tt></a>
completely fails.</td>
<td><a class="reference external" href="../auto_examples/covariance/plot_outlier_detection.html"><img alt="outlier3" src="../_images/plot_outlier_detection_003.png" style="width: 500.0px; height: 250.0px;" /></a></td>
</tr>
</tbody>
</table>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li>See <a class="reference internal" href="../auto_examples/covariance/plot_outlier_detection.html#example-covariance-plot-outlier-detection-py"><em>Outlier detection with several methods.</em></a> for a
comparison of the <a class="reference internal" href="generated/sklearn.svm.OneClassSVM.html#sklearn.svm.OneClassSVM" title="sklearn.svm.OneClassSVM"><tt class="xref py py-class docutils literal"><span class="pre">svm.OneClassSVM</span></tt></a> (tuned to perform like
an outlier detection method) and a covariance-based outlier
detection with <a class="reference internal" href="generated/sklearn.covariance.MinCovDet.html#sklearn.covariance.MinCovDet" title="sklearn.covariance.MinCovDet"><tt class="xref py py-class docutils literal"><span class="pre">covariance.MinCovDet</span></tt></a>.</li>
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