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.
# You should have received a copy of the GNU Lesser General Public License
# 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
"""The Gaussian mixture algorithm for clustering.

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 of 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 :math:`\\Sigma_i`.
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 division, unicode_literals

import logging
from typing import Iterable, NoReturn, Optional, Union

from numpy import ndarray
from sklearn.mixture import GaussianMixture as SKLGaussianMixture

from gemseo.core.dataset import Dataset
from gemseo.mlearning.cluster.cluster import MLPredictiveClusteringAlgo
from gemseo.mlearning.core.ml_algo import TransformerType

LOGGER = logging.getLogger(__name__)

[docs]class GaussianMixture(MLPredictiveClusteringAlgo): """The Gaussian mixture clustering algorithm.""" ABBR = "GaussMix" def __init__( self, data, # type:Dataset transformer=None, # type: Optional[TransformerType] var_names=None, # type: Optional[Iterable[str]] n_components=5, # type: int **parameters # type: Optional[Union[int,float,str,bool]] ): # type: (...) -> None """ Args: n_components: The number of components of the Gaussian mixture. """ 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, # type: ndarray ): # type: (...) -> NoReturn self.labels = self.algo.predict(data) def _predict( self, data, # type: ndarray ): # type: (...) -> ndarray return self.algo.predict(data) def _predict_proba_soft( self, data, # type: ndarray ): # type: (...)-> ndarray return self.algo.predict_proba(data)