Source code for gemseo.mlearning.cluster.gaussian_mixture

# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry,
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# Lesser General Public License for more details.
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# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.

# Contributors:
#    INITIAL AUTHORS - initial API and implementation and/or initial
#                         documentation
#        :author: Syver Doving Agdestein
Gaussian mixture clustering algorithm

The Gaussian mixture algorithm groups the data into clusters.
The number of clusters is fixed. Each cluster :math:`i=1, \\cdots, k` is
defined by a mean :math:`\\mu_i` and a covariance matrix :math:`\\Sigma_i`.

The prediction of the cluster value of a point is simply the cluster where the
probability density from the Gaussian distribution defined by the given mean
and covariance matrix is the highest:

.. math::

        \\operatorname{cluster}(x) =
            \\mathcal{N}(x; \\mu_i, \\Sigma_i) =

where :math:`\\mathcal{N}(x; \\mu_i, \\Sigma_i)` is the value of the
probability density function of a Gaussian random variable
:math:`X \\sim \\mathcal{N}(\\mu_i, \\Sigma_i)` at the point :math:`x` and
:math:`\\|x-\\mu_i\\|_{\\Sigma_i^{-1}} =
\\sqrt{(x-\\mu_i)^T \\Sigma_i^{-1} (x-\\mu_i)}`
is the Mahalanobis distance between :math:`x` and :math:`\\mu_i` weighted by
Likewise, the probability of belonging to a cluster :math:`i=1, \\cdots, k` may
be determined through

.. math::

        \\mathbb{P}(x \\in C_i) = \\frac{\\mathcal{N}(x; \\mu_i, \\Sigma_i)}
            {\\sum_{j=1}^k \\mathcal{N}(x; \\mu_j, \\Sigma_j)},

where :math:`C_i = \\{x\\, | \\, \\operatorname{cluster}(x) = i \\}`.

When fitting the algorithm, the cluster centers :math:`\\mu_i` and the
covariance matrices :math:`\\Sigma_i` are computed using the
expectation-maximization algorithm.

This concept is implemented through the :class:`.GaussianMixture`
class which inherits from the :class:`.MLClusteringAlgo` class.

This clustering algorithm relies on the GaussianMixture class
of the `scikit-learn library <
from __future__ import absolute_import, division, unicode_literals

from future import standard_library
from sklearn.mixture import GaussianMixture as SKLGaussianMixture

from gemseo.mlearning.cluster.cluster import MLClusteringAlgo


from gemseo import LOGGER

[docs]class GaussianMixture(MLClusteringAlgo): """ Gaussian mixture clustering algorithm. """ ABBR = "GaussMix" def __init__( self, data, transformer=None, var_names=None, n_components=5, **parameters ): """Constructor. :param data: learning dataset. :type data: Dataset :param transformer: transformation strategy for data groups. If None, do not transform data. Default: None. :type transformer: dict(str) :param var_names: names of the variables to consider. :type var_names: list(str) :param n_components: number of Gaussian mixture components. Default: 5. :type n_components: int :param parameters: Scikit-learn algorithm parameters. """ super(GaussianMixture, self).__init__( data, transformer=transformer, var_names=var_names, n_components=n_components, **parameters ) self.algo = SKLGaussianMixture(n_components, **parameters) def _fit(self, data): """Fit the clustering model to the data and store labels. :param ndarray data: training data (2D). """ self.labels = self.algo.predict(data) def _predict(self, data): """Predict cluster of data. :param ndarray data: data (2D). :return: clusters of data (1D). :rtype: ndarray(int). """ return self.algo.predict(data) def _predict_proba_soft(self, data): """Predict probability of belonging to each cluster. :param ndarray data: data (2D). :return: probabilities for each cluster for each sample (2D). The sum of each row is one. :rtype: ndarray. """ return self.algo.predict_proba(data)