# encoding: utf-8
# pylint: disable=no-member
# pylint: disable=invalid-name
# pylint: disable=too-many-arguments
"""
This module contains functionality needed for fitting and scoring Gaussian
Mixture Models (GMMs) (needed e.g. in madmom.features.beats).
The needed functionality is taken from sklearn.mixture.GMM which is released
under the BSD license and was written by these authors:
- Ron Weiss <ronweiss@gmail.com>
- Fabian Pedregosa <fabian.pedregosa@inria.fr>
- Bertrand Thirion <bertrand.thirion@inria.fr>
This version works with sklearn v0.16 (and hopefully onwards).
All commits until 0650d5502e01e6b4245ce99729fc8e7a71aacff3 are incorporated.
"""
from __future__ import absolute_import, division, print_function
import numpy as np
from scipy import linalg
# the following code is copied from sklearn
[docs]def logsumexp(arr, axis=0):
"""
Computes the sum of arr assuming arr is in the log domain.
Parameters
----------
arr : numpy array
Input data [log domain].
axis : int, optional
Axis to operate on.
Returns
-------
numpy array
log(sum(exp(arr))) while minimizing the possibility of over/underflow.
Notes
-----
Function copied from sklearn.utils.extmath.
"""
arr = np.rollaxis(arr, axis)
# Use the max to normalize, as with the log this is what accumulates
# the less errors
vmax = arr.max(axis=0)
out = np.log(np.sum(np.exp(arr - vmax), axis=0))
out += vmax
return out
[docs]def pinvh(a, cond=None, rcond=None, lower=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.
Calculate a generalized inverse of a symmetric matrix using its
eigenvalue decomposition and including all 'large' eigenvalues.
Parameters
----------
a : array, shape (N, N)
Real symmetric or complex hermetian matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' eigenvalues. Singular values smaller than rcond *
largest_eigenvalue are considered zero.
If None or -1, suitable machine precision is used.
lower : boolean
Whether the pertinent array data is taken from the lower or upper
triangle of `a`.
Returns
-------
B : array, shape (N, N)
Raises
------
LinAlgError
If eigenvalue does not converge
Notes
-----
Function copied from sklearn.utils.extmath.
"""
a = np.asarray_chkfinite(a)
s, u = linalg.eigh(a, lower=lower)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# unlike svd case, eigh can lead to negative eigenvalues
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = np.zeros_like(s)
psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]
return np.dot(u * psigma_diag, np.conjugate(u).T)
[docs]def log_multivariate_normal_density(x, means, covars, covariance_type='diag'):
"""
Compute the log probability under a multivariate Gaussian distribution.
Parameters
----------
x : array_like, shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds to a
single data point.
means : array_like, shape (n_components, n_features)
List of n_features-dimensional mean vectors for n_components Gaussians.
Each row corresponds to a single mean vector.
covars : array_like
List of n_components covariance parameters for each Gaussian. The shape
depends on `covariance_type`:
- (n_components, n_features) if 'spherical',
- (n_features, n_features) if 'tied',
- (n_components, n_features) if 'diag',
- (n_components, n_features, n_features) if 'full'.
covariance_type : {'diag', 'spherical', 'tied', 'full'}
Type of the covariance parameters. Defaults to 'diag'.
Returns
-------
lpr : array_like, shape (n_samples, n_components)
Array containing the log probabilities of each data point in `x`
under each of the n_components multivariate Gaussian distributions.
"""
log_multivariate_normal_density_dict = {
'spherical': _log_multivariate_normal_density_spherical,
'tied': _log_multivariate_normal_density_tied,
'diag': _log_multivariate_normal_density_diag,
'full': _log_multivariate_normal_density_full}
return log_multivariate_normal_density_dict[covariance_type](
x, means, covars)
def _log_multivariate_normal_density_diag(x, means, covars):
"""Compute Gaussian log-density at x for a diagonal model."""
_, n_dim = x.shape
lpr = -0.5 * (n_dim * np.log(2 * np.pi) + np.sum(np.log(covars), 1) +
np.sum((means ** 2) / covars, 1) -
2 * np.dot(x, (means / covars).T) +
np.dot(x ** 2, (1.0 / covars).T))
return lpr
def _log_multivariate_normal_density_spherical(x, means, covars):
"""Compute Gaussian log-density at x for a spherical model."""
cv = covars.copy()
if covars.ndim == 1:
cv = cv[:, np.newaxis]
if covars.shape[1] == 1:
cv = np.tile(cv, (1, x.shape[-1]))
return _log_multivariate_normal_density_diag(x, means, cv)
def _log_multivariate_normal_density_tied(x, means, covars):
"""Compute Gaussian log-density at X for a tied model"""
cv = np.tile(covars, (means.shape[0], 1, 1))
return _log_multivariate_normal_density_full(x, means, cv)
def _log_multivariate_normal_density_full(x, means, covars, min_covar=1.e-7):
"""Log probability for full covariance matrices."""
n_samples, n_dim = x.shape
nmix = len(means)
log_prob = np.empty((n_samples, nmix))
for c, (mu, cv) in enumerate(zip(means, covars)):
try:
cv_chol = linalg.cholesky(cv, lower=True)
except linalg.LinAlgError:
# The model is most probably stuck in a component with too
# few observations, we need to reinitialize this components
try:
cv_chol = linalg.cholesky(cv + min_covar * np.eye(n_dim),
lower=True)
except linalg.LinAlgError:
raise ValueError("'covars' must be symmetric, "
"positive-definite")
cv_log_det = 2 * np.sum(np.log(np.diagonal(cv_chol)))
cv_sol = linalg.solve_triangular(cv_chol, (x - mu).T, lower=True).T
log_prob[:, c] = - .5 * (np.sum(cv_sol ** 2, axis=1) +
n_dim * np.log(2 * np.pi) + cv_log_det)
return log_prob
# provide a basic GMM implementation which has the score() method implemented
[docs]class GMM(object):
"""
Gaussian Mixture Model
Representation of a Gaussian mixture model probability distribution.
This class allows for easy evaluation of, sampling from, and
maximum-likelihood estimation of the parameters of a GMM distribution.
Initializes parameters such that every mixture component has zero
mean and identity covariance.
Parameters
----------
n_components : int, optional
Number of mixture components. Defaults to 1.
covariance_type : {'diag', 'spherical', 'tied', 'full'}
String describing the type of covariance parameters to
use. Defaults to 'diag'.
Attributes
----------
`weights_` : array, shape (n_components,)
This attribute stores the mixing weights for each mixture component.
`means_` : array, shape (n_components, n_features)
Mean parameters for each mixture component.
`covars_` : array
Covariance parameters for each mixture component. The shape
depends on `covariance_type`.::
- (n_components, n_features) 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.
See Also
--------
sklearn.mixture.GMM
"""
def __init__(self, n_components=1, covariance_type='full'):
if covariance_type not in ['spherical', 'tied', 'diag', 'full']:
raise ValueError('Invalid value for covariance_type: %s' %
covariance_type)
# save parameters
self.n_components = n_components
self.covariance_type = covariance_type
# init variables
self.weights_ = np.ones(self.n_components) / self.n_components
self.means_ = None
self.covars_ = None
self.converged_ = False
[docs] def score_samples(self, x):
"""
Return the per-sample likelihood of the data under the model.
Compute the log probability of x under the model and
return the posterior distribution (responsibilities) of each
mixture component for each element of x.
Parameters
----------
x: array_like, shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds
to a single data point.
Returns
-------
log_prob : array_like, shape (n_samples,)
Log probabilities of each data point in `x`.
responsibilities : array_like, shape (n_samples, n_components)
Posterior probabilities of each mixture component for each
observation.
"""
x = np.asarray(x)
if x.ndim == 1:
x = x[:, np.newaxis]
if x.size == 0:
return np.array([]), np.empty((0, self.n_components))
if x.shape[1] != self.means_.shape[1]:
raise ValueError('The shape of x is not compatible with self')
lpr = (log_multivariate_normal_density(x, self.means_, self.covars_,
self.covariance_type) +
np.log(self.weights_))
log_prob = logsumexp(lpr, axis=1)
responsibilities = np.exp(lpr - log_prob[:, np.newaxis])
return log_prob, responsibilities
[docs] def score(self, x):
"""
Compute the log probability under the model.
Parameters
----------
x : array_like, shape (n_samples, n_features)
List of n_features-dimensional data points. Each row
corresponds to a single data point.
Returns
-------
log_prob : array_like, shape (n_samples,)
Log probabilities of each data point in `x`.
"""
log_prob, _ = self.score_samples(x)
return log_prob
[docs] def fit(self, x, random_state=None, tol=1e-3, min_covar=1e-3,
n_iter=100, n_init=1, params='wmc', init_params='wmc'):
"""
Estimate model parameters with the expectation-maximization algorithm.
A initialization step is performed before entering the em
algorithm. If you want to avoid this step, set the keyword
argument init_params to the empty string '' when creating the
GMM object. Likewise, if you would like just to do an
initialization, set n_iter=0.
Parameters
----------
x : array_like, shape (n, n_features)
List of n_features-dimensional data points. Each row corresponds
to a single data point.
random_state: RandomState or an int seed (0 by default)
A random number generator instance.
min_covar : float, optional
Floor on the diagonal of the covariance matrix to prevent
overfitting.
tol : float, optional
Convergence threshold. EM iterations will stop when average
gain in log-likelihood is below this threshold.
n_iter : int, optional
Number of EM iterations to perform.
n_init : int, optional
Number of initializations to perform, the best results is kept.
params : str, optional
Controls which parameters are updated in the training process.
Can contain any combination of 'w' for weights, 'm' for means,
and 'c' for covars.
init_params : str, optional
Controls which parameters are updated in the initialization
process. Can contain any combination of 'w' for weights,
'm' for means, and 'c' for covars.
"""
import sklearn.mixture
# first initialise a sklearn.mixture.GMM object
gmm = sklearn.mixture.GMM(n_components=self.n_components,
covariance_type=self.covariance_type,
random_state=random_state, tol=tol,
min_covar=min_covar, n_iter=n_iter,
n_init=n_init, params=params,
init_params=init_params)
# fit this GMM
gmm.fit(x)
# copy the needed information
self.converged_ = gmm.converged_
self.covars_ = gmm.covars_
self.means_ = gmm.means_
self.weights_ = gmm.weights_
# and return self
return self