.. _outlier_detection:

===================================================
Novelty and Outlier Detection
===================================================

.. currentmodule:: sklearn

Many applications require being able to decide whether a new observation
belongs to the same distribution as existing observations (it is an
`inlier`), 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:

:novelty detection: 
  The training data is not polluted by outliers, and we are interested in
  detecting anomalies in new observations.

:outlier detection:
  The training data contains outliers, and we need to fit the central
  mode of the training data, ignoring the deviant observations.

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::

    estimator.fit(X_train)

new observations can then be sorted as inliers or outliers with a
`predict` method::

    estimator.predict(X_test)

Inliers are labeled 1, while outliers are labeled -1.

Novelty Detection
=================

Consider a data set of :math:`n` observations from the same
distribution described by :math:`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.

In general, it is about to learn a rough, close frontier delimiting
the contour of the initial observations distribution, plotted in
embedding :math:`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.

The One-Class SVM has been introduced by Schölkopf et al. for that purpose 
and implemented in the :ref:`svm` module in the
:class:`svm.OneClassSVM` 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 :math:`\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.

.. topic:: References:

    * `Estimating the support of a high-dimensional distribution
      <http://dl.acm.org/citation.cfm?id=1119749>`_ Schölkopf, 
      Bernhard, et al. Neural computation 13.7 (2001): 1443-1471.
      
.. topic:: Examples:

   * See :ref:`example_svm_plot_oneclass.py` for visualizing the
     frontier learned around some data by a
     :class:`svm.OneClassSVM` object.

.. figure:: ../auto_examples/svm/images/plot_oneclass_001.png
   :target: ../auto_examples/svm/plot_oneclasse.html
   :align: center
   :scale: 75%
   

Outlier Detection
=================

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.


Fitting an elliptic envelope
----------------------------

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. 

The scikit-learn provides an object
:class:`covariance.EllipticEnvelope` 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.

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.

.. figure:: ../auto_examples/covariance/images/plot_mahalanobis_distances_001.png
   :target: ../auto_examples/covariance/plot_mahalanobis_distances.html
   :align: center
   :scale: 75%

.. topic:: Examples:

   * See :ref:`example_covariance_plot_mahalanobis_distances.py` for
     an illustration of the difference between using a standard
     (:class:`covariance.EmpiricalCovariance`) or a robust estimate
     (:class:`covariance.MinCovDet`) of location and covariance to
     assess the degree of outlyingness of an observation.

.. topic:: References:

    ..  [RD1999] Rousseeuw, P.J., Van Driessen, K. "A fast algorithm for the minimum
        covariance determinant estimator" Technometrics 41(3), 212 (1999)
     
One-class SVM versus elliptic envelope
--------------------------------------

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.

The examples below illustrate how the performance of the
:class:`covariance.EllipticEnvelope` degrades as the data is less and
less unimodal.  :class:`svm.OneClassSVM` works better on data with
multiple modes.

.. |outlier1| image:: ../auto_examples/covariance/images/plot_outlier_detection_001.png
   :target: ../auto_examples/covariance/plot_outlier_detection.html
   :scale: 50%

.. |outlier2| image:: ../auto_examples/covariance/images/plot_outlier_detection_002.png
   :target: ../auto_examples/covariance/plot_outlier_detection.html
   :scale: 50%

.. |outlier3| image:: ../auto_examples/covariance/images/plot_outlier_detection_003.png
   :target: ../auto_examples/covariance/plot_outlier_detection.html
   :scale: 50%

.. list-table:: **Comparing One-class SVM approach, and elliptic envelope**
   :widths: 40 60

   *
      - For a inlier mode well-centered and elliptic, the
        :class:`svm.OneClassSVM` 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
        :class:`covariance.EllipticEnvelope` learns an ellipse, which
        fits well the inlier distribution.
      - |outlier1| 

   * 
      - As the inlier distribution becomes bimodal, the
        :class:`covariance.EllipticEnvelope` does not fit well the
        inliers. However, we can see that the :class:`svm.OneClassSVM`
        tends to overfit: because it has not model of inliers, it
        interprets a region where, by chance some outliers are
        clustered, as inliers.
      - |outlier2| 

   * 
      - If the inlier distribution is strongly non Gaussian, the
        :class:`svm.OneClassSVM` is able to recover a reasonable
        approximation, whereas the :class:`covariance.EllipticEnvelope`
        completely fails.
      - |outlier3|

.. topic:: Examples:

   * See :ref:`example_covariance_plot_outlier_detection.py` for a
     comparison of the :class:`svm.OneClassSVM` (tuned to perform like
     an outlier detection method) and a covariance-based outlier
     detection with :class:`covariance.MinCovDet`.


