"""Gaussian Mixture Model."""
# Author: Wei Xue <xuewei4d@gmail.com>
# Modified by Thierry Guillemot <thierry.guillemot.work@gmail.com>
# License: BSD 3 clause
import numpy as np
from scipy import linalg
from .base import BaseMixture, _check_shape
from ..externals.six.moves import zip
from ..utils import check_array
from ..utils.validation import check_is_fitted
from ..utils.extmath import row_norms
###############################################################################
# Gaussian mixture shape checkers used by the GaussianMixture class
def _check_weights(weights, n_components):
"""Check the user provided 'weights'.
Parameters
----------
weights : array-like, shape (n_components,)
The proportions of components of each mixture.
n_components : int
Number of components.
Returns
-------
weights : array, shape (n_components,)
"""
weights = check_array(weights, dtype=[np.float64, np.float32],
ensure_2d=False)
_check_shape(weights, (n_components,), 'weights')
# check range
if (any(np.less(weights, 0.)) or
any(np.greater(weights, 1.))):
raise ValueError("The parameter 'weights' should be in the range "
"[0, 1], but got max value %.5f, min value %.5f"
% (np.min(weights), np.max(weights)))
# check normalization
if not np.allclose(np.abs(1. - np.sum(weights)), 0.):
raise ValueError("The parameter 'weights' should be normalized, "
"but got sum(weights) = %.5f" % np.sum(weights))
return weights
def _check_means(means, n_components, n_features):
"""Validate the provided 'means'.
Parameters
----------
means : array-like, shape (n_components, n_features)
The centers of the current components.
n_components : int
Number of components.
n_features : int
Number of features.
Returns
-------
means : array, (n_components, n_features)
"""
means = check_array(means, dtype=[np.float64, np.float32], ensure_2d=False)
_check_shape(means, (n_components, n_features), 'means')
return means
def _check_precision_positivity(precision, covariance_type):
"""Check a precision vector is positive-definite."""
if np.any(np.less_equal(precision, 0.0)):
raise ValueError("'%s precision' should be "
"positive" % covariance_type)
def _check_precision_matrix(precision, covariance_type):
"""Check a precision matrix is symmetric and positive-definite."""
if not (np.allclose(precision, precision.T) and
np.all(linalg.eigvalsh(precision) > 0.)):
raise ValueError("'%s precision' should be symmetric, "
"positive-definite" % covariance_type)
def _check_precisions_full(precisions, covariance_type):
"""Check the precision matrices are symmetric and positive-definite."""
for prec in precisions:
_check_precision_matrix(prec, covariance_type)
def _check_precisions(precisions, covariance_type, n_components, n_features):
"""Validate user provided precisions.
Parameters
----------
precisions : array-like,
'full' : shape of (n_components, n_features, n_features)
'tied' : shape of (n_features, n_features)
'diag' : shape of (n_components, n_features)
'spherical' : shape of (n_components,)
covariance_type : string
n_components : int
Number of components.
n_features : int
Number of features.
Returns
-------
precisions : array
"""
precisions = check_array(precisions, dtype=[np.float64, np.float32],
ensure_2d=False,
allow_nd=covariance_type == 'full')
precisions_shape = {'full': (n_components, n_features, n_features),
'tied': (n_features, n_features),
'diag': (n_components, n_features),
'spherical': (n_components,)}
_check_shape(precisions, precisions_shape[covariance_type],
'%s precision' % covariance_type)
_check_precisions = {'full': _check_precisions_full,
'tied': _check_precision_matrix,
'diag': _check_precision_positivity,
'spherical': _check_precision_positivity}
_check_precisions[covariance_type](precisions, covariance_type)
return precisions
###############################################################################
# Gaussian mixture parameters estimators (used by the M-Step)
def _estimate_gaussian_covariances_full(resp, X, nk, means, reg_covar):
"""Estimate the full covariance matrices.
Parameters
----------
resp : array-like, shape (n_samples, n_components)
X : array-like, shape (n_samples, n_features)
nk : array-like, shape (n_components,)
means : array-like, shape (n_components, n_features)
reg_covar : float
Returns
-------
covariances : array, shape (n_components, n_features, n_features)
The covariance matrix of the current components.
"""
n_components, n_features = means.shape
covariances = np.empty((n_components, n_features, n_features))
for k in range(n_components):
diff = X - means[k]
covariances[k] = np.dot(resp[:, k] * diff.T, diff) / nk[k]
covariances[k].flat[::n_features + 1] += reg_covar
return covariances
def _estimate_gaussian_covariances_tied(resp, X, nk, means, reg_covar):
"""Estimate the tied covariance matrix.
Parameters
----------
resp : array-like, shape (n_samples, n_components)
X : array-like, shape (n_samples, n_features)
nk : array-like, shape (n_components,)
means : array-like, shape (n_components, n_features)
reg_covar : float
Returns
-------
covariance : array, shape (n_features, n_features)
The tied covariance matrix of the components.
"""
avg_X2 = np.dot(X.T, X)
avg_means2 = np.dot(nk * means.T, means)
covariance = avg_X2 - avg_means2
covariance /= nk.sum()
covariance.flat[::len(covariance) + 1] += reg_covar
return covariance
def _estimate_gaussian_covariances_diag(resp, X, nk, means, reg_covar):
"""Estimate the diagonal covariance vectors.
Parameters
----------
responsibilities : array-like, shape (n_samples, n_components)
X : array-like, shape (n_samples, n_features)
nk : array-like, shape (n_components,)
means : array-like, shape (n_components, n_features)
reg_covar : float
Returns
-------
covariances : array, shape (n_components, n_features)
The covariance vector of the current components.
"""
avg_X2 = np.dot(resp.T, X * X) / nk[:, np.newaxis]
avg_means2 = means ** 2
avg_X_means = means * np.dot(resp.T, X) / nk[:, np.newaxis]
return avg_X2 - 2 * avg_X_means + avg_means2 + reg_covar
def _estimate_gaussian_covariances_spherical(resp, X, nk, means, reg_covar):
"""Estimate the spherical variance values.
Parameters
----------
responsibilities : array-like, shape (n_samples, n_components)
X : array-like, shape (n_samples, n_features)
nk : array-like, shape (n_components,)
means : array-like, shape (n_components, n_features)
reg_covar : float
Returns
-------
variances : array, shape (n_components,)
The variance values of each components.
"""
return _estimate_gaussian_covariances_diag(resp, X, nk,
means, reg_covar).mean(1)
def _estimate_gaussian_parameters(X, resp, reg_covar, covariance_type):
"""Estimate the Gaussian distribution parameters.
Parameters
----------
X : array-like, shape (n_samples, n_features)
The input data array.
resp : array-like, shape (n_samples, n_components)
The responsibilities for each data sample in X.
reg_covar : float
The regularization added to the diagonal of the covariance matrices.
covariance_type : {'full', 'tied', 'diag', 'spherical'}
The type of precision matrices.
Returns
-------
nk : array-like, shape (n_components,)
The numbers of data samples in the current components.
means : array-like, shape (n_components, n_features)
The centers of the current components.
covariances : array-like
The covariance matrix of the current components.
The shape depends of the covariance_type.
"""
nk = resp.sum(axis=0) + 10 * np.finfo(resp.dtype).eps
means = np.dot(resp.T, X) / nk[:, np.newaxis]
covariances = {"full": _estimate_gaussian_covariances_full,
"tied": _estimate_gaussian_covariances_tied,
"diag": _estimate_gaussian_covariances_diag,
"spherical": _estimate_gaussian_covariances_spherical
}[covariance_type](resp, X, nk, means, reg_covar)
return nk, means, covariances
def _compute_precision_cholesky(covariances, covariance_type):
"""Compute the Cholesky decomposition of the precisions.
Parameters
----------
covariances : array-like
The covariance matrix of the current components.
The shape depends of the covariance_type.
covariance_type : {'full', 'tied', 'diag', 'spherical'}
The type of precision matrices.
Returns
-------
precisions_cholesky : array-like
The cholesky decomposition of sample precisions of the current
components. The shape depends of the covariance_type.
"""
estimate_precision_error_message = (
"Fitting the mixture model failed because some components have "
"ill-defined empirical covariance (for instance caused by singleton "
"or collapsed samples). Try to decrease the number of components, "
"or increase reg_covar.")
if covariance_type in 'full':
n_components, n_features, _ = covariances.shape
precisions_chol = np.empty((n_components, n_features, n_features))
for k, covariance in enumerate(covariances):
try:
cov_chol = linalg.cholesky(covariance, lower=True)
except linalg.LinAlgError:
raise ValueError(estimate_precision_error_message)
precisions_chol[k] = linalg.solve_triangular(cov_chol,
np.eye(n_features),
lower=True).T
elif covariance_type == 'tied':
_, n_features = covariances.shape
try:
cov_chol = linalg.cholesky(covariances, lower=True)
except linalg.LinAlgError:
raise ValueError(estimate_precision_error_message)
precisions_chol = linalg.solve_triangular(cov_chol, np.eye(n_features),
lower=True).T
else:
if np.any(np.less_equal(covariances, 0.0)):
raise ValueError(estimate_precision_error_message)
precisions_chol = 1. / np.sqrt(covariances)
return precisions_chol
###############################################################################
# Gaussian mixture probability estimators
def _compute_log_det_cholesky(matrix_chol, covariance_type, n_features):
"""Compute the log-det of the cholesky decomposition of matrices.
Parameters
----------
matrix_chol : array-like,
Cholesky decompositions of the matrices.
'full' : shape of (n_components, n_features, n_features)
'tied' : shape of (n_features, n_features)
'diag' : shape of (n_components, n_features)
'spherical' : shape of (n_components,)
covariance_type : {'full', 'tied', 'diag', 'spherical'}
n_features : int
Number of features.
Returns
-------
log_det_precision_chol : array-like, shape (n_components,)
The determinant of the precision matrix for each component.
"""
if covariance_type == 'full':
n_components, _, _ = matrix_chol.shape
log_det_chol = (np.sum(np.log(
matrix_chol.reshape(
n_components, -1)[:, ::n_features + 1]), 1))
elif covariance_type == 'tied':
log_det_chol = (np.sum(np.log(np.diag(matrix_chol))))
elif covariance_type == 'diag':
log_det_chol = (np.sum(np.log(matrix_chol), axis=1))
else:
log_det_chol = n_features * (np.log(matrix_chol))
return log_det_chol
def _estimate_log_gaussian_prob(X, means, precisions_chol, covariance_type):
"""Estimate the log Gaussian probability.
Parameters
----------
X : array-like, shape (n_samples, n_features)
means : array-like, shape (n_components, n_features)
precisions_chol : array-like,
Cholesky decompositions of the precision matrices.
'full' : shape of (n_components, n_features, n_features)
'tied' : shape of (n_features, n_features)
'diag' : shape of (n_components, n_features)
'spherical' : shape of (n_components,)
covariance_type : {'full', 'tied', 'diag', 'spherical'}
Returns
-------
log_prob : array, shape (n_samples, n_components)
"""
n_samples, n_features = X.shape
n_components, _ = means.shape
# det(precision_chol) is half of det(precision)
log_det = _compute_log_det_cholesky(
precisions_chol, covariance_type, n_features)
if covariance_type == 'full':
log_prob = np.empty((n_samples, n_components))
for k, (mu, prec_chol) in enumerate(zip(means, precisions_chol)):
y = np.dot(X, prec_chol) - np.dot(mu, prec_chol)
log_prob[:, k] = np.sum(np.square(y), axis=1)
elif covariance_type == 'tied':
log_prob = np.empty((n_samples, n_components))
for k, mu in enumerate(means):
y = np.dot(X, precisions_chol) - np.dot(mu, precisions_chol)
log_prob[:, k] = np.sum(np.square(y), axis=1)
elif covariance_type == 'diag':
precisions = precisions_chol ** 2
log_prob = (np.sum((means ** 2 * precisions), 1) -
2. * np.dot(X, (means * precisions).T) +
np.dot(X ** 2, precisions.T))
elif covariance_type == 'spherical':
precisions = precisions_chol ** 2
log_prob = (np.sum(means ** 2, 1) * precisions -
2 * np.dot(X, means.T * precisions) +
np.outer(row_norms(X, squared=True), precisions))
return -.5 * (n_features * np.log(2 * np.pi) + log_prob) + log_det
class GaussianMixture(BaseMixture):
"""Gaussian Mixture.
Representation of a Gaussian mixture model probability distribution.
This class allows to estimate the parameters of a Gaussian mixture
distribution.
Read more in the :ref:`User Guide <gmm>`.
.. versionadded:: 0.18
Parameters
----------
n_components : int, defaults to 1.
The number of mixture components.
covariance_type : {'full', 'tied', 'diag', 'spherical'},
defaults to 'full'.
String describing the type of covariance parameters to use.
Must be one of::
'full' (each component has its own general covariance matrix),
'tied' (all components share the same general covariance matrix),
'diag' (each component has its own diagonal covariance matrix),
'spherical' (each component has its own single variance).
tol : float, defaults to 1e-3.
The convergence threshold. EM iterations will stop when the
lower bound average gain is below this threshold.
reg_covar : float, defaults to 1e-6.
Non-negative regularization added to the diagonal of covariance.
Allows to assure that the covariance matrices are all positive.
max_iter : int, defaults to 100.
The number of EM iterations to perform.
n_init : int, defaults to 1.
The number of initializations to perform. The best results are kept.
init_params : {'kmeans', 'random'}, defaults to 'kmeans'.
The method used to initialize the weights, the means and the
precisions.
Must be one of::
'kmeans' : responsibilities are initialized using kmeans.
'random' : responsibilities are initialized randomly.
weights_init : array-like, shape (n_components, ), optional
The user-provided initial weights, defaults to None.
If it None, weights are initialized using the `init_params` method.
means_init : array-like, shape (n_components, n_features), optional
The user-provided initial means, defaults to None,
If it None, means are initialized using the `init_params` method.
precisions_init : array-like, optional.
The user-provided initial precisions (inverse of the covariance
matrices), defaults to None.
If it None, precisions are initialized using the 'init_params' method.
The shape depends on 'covariance_type'::
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
random_state : int, RandomState instance or None, optional (default=None)
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
warm_start : bool, default to False.
If 'warm_start' is True, the solution of the last fitting is used as
initialization for the next call of fit(). This can speed up
convergence when fit is called several time on similar problems.
verbose : int, default to 0.
Enable verbose output. If 1 then it prints the current
initialization and each iteration step. If greater than 1 then
it prints also the log probability and the time needed
for each step.
verbose_interval : int, default to 10.
Number of iteration done before the next print.
Attributes
----------
weights_ : array-like, shape (n_components,)
The weights of each mixture components.
means_ : array-like, shape (n_components, n_features)
The mean of each mixture component.
covariances_ : array-like
The covariance of each mixture component.
The shape depends on `covariance_type`::
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
precisions_ : array-like
The precision matrices for each component in the mixture. A precision
matrix is the inverse of a covariance matrix. A covariance matrix is
symmetric positive definite so the mixture of Gaussian can be
equivalently parameterized by the precision matrices. Storing the
precision matrices instead of the covariance matrices makes it more
efficient to compute the log-likelihood of new samples at test time.
The shape depends on `covariance_type`::
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
precisions_cholesky_ : array-like
The cholesky decomposition of the precision matrices of each mixture
component. A precision matrix is the inverse of a covariance matrix.
A covariance matrix is symmetric positive definite so the mixture of
Gaussian can be equivalently parameterized by the precision matrices.
Storing the precision matrices instead of the covariance matrices makes
it more efficient to compute the log-likelihood of new samples at test
time. The shape depends on `covariance_type`::
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
converged_ : bool
True when convergence was reached in fit(), False otherwise.
n_iter_ : int
Number of step used by the best fit of EM to reach the convergence.
lower_bound_ : float
Log-likelihood of the best fit of EM.
See Also
--------
BayesianGaussianMixture : Gaussian mixture model fit with a variational
inference.
"""
def __init__(self, n_components=1, covariance_type='full', tol=1e-3,
reg_covar=1e-6, max_iter=100, n_init=1, init_params='kmeans',
weights_init=None, means_init=None, precisions_init=None,
random_state=None, warm_start=False,
verbose=0, verbose_interval=10):
super(GaussianMixture, self).__init__(
n_components=n_components, tol=tol, reg_covar=reg_covar,
max_iter=max_iter, n_init=n_init, init_params=init_params,
random_state=random_state, warm_start=warm_start,
verbose=verbose, verbose_interval=verbose_interval)
self.covariance_type = covariance_type
self.weights_init = weights_init
self.means_init = means_init
self.precisions_init = precisions_init
def _check_parameters(self, X):
"""Check the Gaussian mixture parameters are well defined."""
_, n_features = X.shape
if self.covariance_type not in ['spherical', 'tied', 'diag', 'full']:
raise ValueError("Invalid value for 'covariance_type': %s "
"'covariance_type' should be in "
"['spherical', 'tied', 'diag', 'full']"
% self.covariance_type)
if self.weights_init is not None:
self.weights_init = _check_weights(self.weights_init,
self.n_components)
if self.means_init is not None:
self.means_init = _check_means(self.means_init,
self.n_components, n_features)
if self.precisions_init is not None:
self.precisions_init = _check_precisions(self.precisions_init,
self.covariance_type,
self.n_components,
n_features)
def _initialize(self, X, resp):
"""Initialization of the Gaussian mixture parameters.
Parameters
----------
X : array-like, shape (n_samples, n_features)
resp : array-like, shape (n_samples, n_components)
"""
n_samples, _ = X.shape
weights, means, covariances = _estimate_gaussian_parameters(
X, resp, self.reg_covar, self.covariance_type)
weights /= n_samples
self.weights_ = (weights if self.weights_init is None
else self.weights_init)
self.means_ = means if self.means_init is None else self.means_init
if self.precisions_init is None:
self.covariances_ = covariances
self.precisions_cholesky_ = _compute_precision_cholesky(
covariances, self.covariance_type)
elif self.covariance_type == 'full':
self.precisions_cholesky_ = np.array(
[linalg.cholesky(prec_init, lower=True)
for prec_init in self.precisions_init])
elif self.covariance_type == 'tied':
self.precisions_cholesky_ = linalg.cholesky(self.precisions_init,
lower=True)
else:
self.precisions_cholesky_ = self.precisions_init
def _m_step(self, X, log_resp):
"""M step.
Parameters
----------
X : array-like, shape (n_samples, n_features)
log_resp : array-like, shape (n_samples, n_components)
Logarithm of the posterior probabilities (or responsibilities) of
the point of each sample in X.
"""
n_samples, _ = X.shape
self.weights_, self.means_, self.covariances_ = (
_estimate_gaussian_parameters(X, np.exp(log_resp), self.reg_covar,
self.covariance_type))
self.weights_ /= n_samples
self.precisions_cholesky_ = _compute_precision_cholesky(
self.covariances_, self.covariance_type)
def _estimate_log_prob(self, X):
return _estimate_log_gaussian_prob(
X, self.means_, self.precisions_cholesky_, self.covariance_type)
def _estimate_log_weights(self):
return np.log(self.weights_)
def _compute_lower_bound(self, _, log_prob_norm):
return log_prob_norm
def _check_is_fitted(self):
check_is_fitted(self, ['weights_', 'means_', 'precisions_cholesky_'])
def _get_parameters(self):
return (self.weights_, self.means_, self.covariances_,
self.precisions_cholesky_)
def _set_parameters(self, params):
(self.weights_, self.means_, self.covariances_,
self.precisions_cholesky_) = params
# Attributes computation
_, n_features = self.means_.shape
if self.covariance_type == 'full':
self.precisions_ = np.empty(self.precisions_cholesky_.shape)
for k, prec_chol in enumerate(self.precisions_cholesky_):
self.precisions_[k] = np.dot(prec_chol, prec_chol.T)
elif self.covariance_type == 'tied':
self.precisions_ = np.dot(self.precisions_cholesky_,
self.precisions_cholesky_.T)
else:
self.precisions_ = self.precisions_cholesky_ ** 2
def _n_parameters(self):
"""Return the number of free parameters in the model."""
_, n_features = self.means_.shape
if self.covariance_type == 'full':
cov_params = self.n_components * n_features * (n_features + 1) / 2.
elif self.covariance_type == 'diag':
cov_params = self.n_components * n_features
elif self.covariance_type == 'tied':
cov_params = n_features * (n_features + 1) / 2.
elif self.covariance_type == 'spherical':
cov_params = self.n_components
mean_params = n_features * self.n_components
return int(cov_params + mean_params + self.n_components - 1)
[docs] def bic(self, X):
"""Bayesian information criterion for the current model on the input X.
Parameters
----------
X : array of shape (n_samples, n_dimensions)
Returns
-------
bic : float
The lower the better.
"""
return (-2 * self.score(X) * X.shape[0] +
self._n_parameters() * np.log(X.shape[0]))
[docs] def aic(self, X):
"""Akaike information criterion for the current model on the input X.
Parameters
----------
X : array of shape (n_samples, n_dimensions)
Returns
-------
aic : float
The lower the better.
"""
return -2 * self.score(X) * X.shape[0] + 2 * self._n_parameters()
