"""Bayesian Gaussian Mixture Model.""" # Author: Wei Xue # Thierry Guillemot # License: BSD 3 clause import math import numpy as np from scipy.special import betaln, digamma, gammaln from .base import BaseMixture, _check_shape from .gaussian_mixture import _check_precision_matrix from .gaussian_mixture import _check_precision_positivity from .gaussian_mixture import _compute_log_det_cholesky from .gaussian_mixture import _compute_precision_cholesky from .gaussian_mixture import _estimate_gaussian_parameters from .gaussian_mixture import _estimate_log_gaussian_prob from ..utils import check_array from ..utils.validation import check_is_fitted def _log_dirichlet_norm(dirichlet_concentration): """Compute the log of the Dirichlet distribution normalization term. Parameters ---------- dirichlet_concentration : array-like, shape (n_samples,) The parameters values of the Dirichlet distribution. Returns ------- log_dirichlet_norm : float The log normalization of the Dirichlet distribution. """ return (gammaln(np.sum(dirichlet_concentration)) - np.sum(gammaln(dirichlet_concentration))) def _log_wishart_norm(degrees_of_freedom, log_det_precisions_chol, n_features): """Compute the log of the Wishart distribution normalization term. Parameters ---------- degrees_of_freedom : array-like, shape (n_components,) The number of degrees of freedom on the covariance Wishart distributions. log_det_precision_chol : array-like, shape (n_components,) The determinant of the precision matrix for each component. n_features : int The number of features. Return ------ log_wishart_norm : array-like, shape (n_components,) The log normalization of the Wishart distribution. """ # To simplify the computation we have removed the np.log(np.pi) term return -(degrees_of_freedom * log_det_precisions_chol + degrees_of_freedom * n_features * .5 * math.log(2.) + np.sum(gammaln(.5 * (degrees_of_freedom - np.arange(n_features)[:, np.newaxis])), 0)) class BayesianGaussianMixture(BaseMixture): """Variational Bayesian estimation of a Gaussian mixture. This class allows to infer an approximate posterior distribution over the parameters of a Gaussian mixture distribution. The effective number of components can be inferred from the data. This class implements two types of prior for the weights distribution: a finite mixture model with Dirichlet distribution and an infinite mixture model with the Dirichlet Process. In practice Dirichlet Process inference algorithm is approximated and uses a truncated distribution with a fixed maximum number of components (called the Stick-breaking representation). The number of components actually used almost always depends on the data. .. versionadded:: 0.18 Read more in the :ref:`User Guide `. Parameters ---------- n_components : int, defaults to 1. The number of mixture components. Depending on the data and the value of the `weight_concentration_prior` the model can decide to not use all the components by setting some component `weights_` to values very close to zero. The number of effective components is therefore smaller than n_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 on the likelihood (of the training data with respect to the model) 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 result with the highest lower bound value on the likelihood is kept. init_params : {'kmeans', 'random'}, defaults to 'kmeans'. The method used to initialize the weights, the means and the covariances. Must be one of:: 'kmeans' : responsibilities are initialized using kmeans. 'random' : responsibilities are initialized randomly. weight_concentration_prior_type : str, defaults to 'dirichlet_process'. String describing the type of the weight concentration prior. Must be one of:: 'dirichlet_process' (using the Stick-breaking representation), 'dirichlet_distribution' (can favor more uniform weights). weight_concentration_prior : float | None, optional. The dirichlet concentration of each component on the weight distribution (Dirichlet). This is commonly called gamma in the literature. The higher concentration puts more mass in the center and will lead to more components being active, while a lower concentration parameter will lead to more mass at the edge of the mixture weights simplex. The value of the parameter must be greater than 0. If it is None, it's set to ``1. / n_components``. mean_precision_prior : float | None, optional. The precision prior on the mean distribution (Gaussian). Controls the extend to where means can be placed. Smaller values concentrate the means of each clusters around `mean_prior`. The value of the parameter must be greater than 0. If it is None, it's set to 1. mean_prior : array-like, shape (n_features,), optional The prior on the mean distribution (Gaussian). If it is None, it's set to the mean of X. degrees_of_freedom_prior : float | None, optional. The prior of the number of degrees of freedom on the covariance distributions (Wishart). If it is None, it's set to `n_features`. covariance_prior : float or array-like, optional The prior on the covariance distribution (Wishart). If it is None, the emiprical covariance prior is initialized using the covariance of X. The shape depends on `covariance_type`:: (n_features, n_features) if 'full', (n_features, n_features) if 'tied', (n_features) if 'diag', float if 'spherical' 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 inference to reach the convergence. lower_bound_ : float Lower bound value on the likelihood (of the training data with respect to the model) of the best fit of inference. weight_concentration_prior_ : tuple or float The dirichlet concentration of each component on the weight distribution (Dirichlet). The type depends on ``weight_concentration_prior_type``:: (float, float) if 'dirichlet_process' (Beta parameters), float if 'dirichlet_distribution' (Dirichlet parameters). The higher concentration puts more mass in the center and will lead to more components being active, while a lower concentration parameter will lead to more mass at the edge of the simplex. weight_concentration_ : array-like, shape (n_components,) The dirichlet concentration of each component on the weight distribution (Dirichlet). mean_precision_prior : float The precision prior on the mean distribution (Gaussian). Controls the extend to where means can be placed. Smaller values concentrate the means of each clusters around `mean_prior`. mean_precision_ : array-like, shape (n_components,) The precision of each components on the mean distribution (Gaussian). means_prior_ : array-like, shape (n_features,) The prior on the mean distribution (Gaussian). degrees_of_freedom_prior_ : float The prior of the number of degrees of freedom on the covariance distributions (Wishart). degrees_of_freedom_ : array-like, shape (n_components,) The number of degrees of freedom of each components in the model. covariance_prior_ : float or array-like The prior on the covariance distribution (Wishart). The shape depends on `covariance_type`:: (n_features, n_features) if 'full', (n_features, n_features) if 'tied', (n_features) if 'diag', float if 'spherical' See Also -------- GaussianMixture : Finite Gaussian mixture fit with EM. References ---------- .. [1] `Bishop, Christopher M. (2006). "Pattern recognition and machine learning". Vol. 4 No. 4. New York: Springer. `_ .. [2] `Hagai Attias. (2000). "A Variational Bayesian Framework for Graphical Models". In Advances in Neural Information Processing Systems 12. `_ .. [3] `Blei, David M. and Michael I. Jordan. (2006). "Variational inference for Dirichlet process mixtures". Bayesian analysis 1.1 `_ """ 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', weight_concentration_prior_type='dirichlet_process', weight_concentration_prior=None, mean_precision_prior=None, mean_prior=None, degrees_of_freedom_prior=None, covariance_prior=None, random_state=None, warm_start=False, verbose=0, verbose_interval=10): super(BayesianGaussianMixture, 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.weight_concentration_prior_type = weight_concentration_prior_type self.weight_concentration_prior = weight_concentration_prior self.mean_precision_prior = mean_precision_prior self.mean_prior = mean_prior self.degrees_of_freedom_prior = degrees_of_freedom_prior self.covariance_prior = covariance_prior def _check_parameters(self, X): """Check that the parameters are well defined. Parameters ---------- X : array-like, shape (n_samples, n_features) """ 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.weight_concentration_prior_type not in ['dirichlet_process', 'dirichlet_distribution']): raise ValueError( "Invalid value for 'weight_concentration_prior_type': %s " "'weight_concentration_prior_type' should be in " "['dirichlet_process', 'dirichlet_distribution']" % self.weight_concentration_prior_type) self._check_weights_parameters() self._check_means_parameters(X) self._check_precision_parameters(X) self._checkcovariance_prior_parameter(X) def _check_weights_parameters(self): """Check the parameter of the Dirichlet distribution.""" if self.weight_concentration_prior is None: self.weight_concentration_prior_ = 1. / self.n_components elif self.weight_concentration_prior > 0.: self.weight_concentration_prior_ = ( self.weight_concentration_prior) else: raise ValueError("The parameter 'weight_concentration_prior' " "should be greater than 0., but got %.3f." % self.weight_concentration_prior) def _check_means_parameters(self, X): """Check the parameters of the Gaussian distribution. Parameters ---------- X : array-like, shape (n_samples, n_features) """ _, n_features = X.shape if self.mean_precision_prior is None: self.mean_precision_prior_ = 1. elif self.mean_precision_prior > 0.: self.mean_precision_prior_ = self.mean_precision_prior else: raise ValueError("The parameter 'mean_precision_prior' should be " "greater than 0., but got %.3f." % self.mean_precision_prior) if self.mean_prior is None: self.mean_prior_ = X.mean(axis=0) else: self.mean_prior_ = check_array(self.mean_prior, dtype=[np.float64, np.float32], ensure_2d=False) _check_shape(self.mean_prior_, (n_features, ), 'means') def _check_precision_parameters(self, X): """Check the prior parameters of the precision distribution. Parameters ---------- X : array-like, shape (n_samples, n_features) """ _, n_features = X.shape if self.degrees_of_freedom_prior is None: self.degrees_of_freedom_prior_ = n_features elif self.degrees_of_freedom_prior > n_features - 1.: self.degrees_of_freedom_prior_ = self.degrees_of_freedom_prior else: raise ValueError("The parameter 'degrees_of_freedom_prior' " "should be greater than %d, but got %.3f." % (n_features - 1, self.degrees_of_freedom_prior)) def _checkcovariance_prior_parameter(self, X): """Check the `covariance_prior_`. Parameters ---------- X : array-like, shape (n_samples, n_features) """ _, n_features = X.shape if self.covariance_prior is None: self.covariance_prior_ = { 'full': np.atleast_2d(np.cov(X.T)), 'tied': np.atleast_2d(np.cov(X.T)), 'diag': np.var(X, axis=0, ddof=1), 'spherical': np.var(X, axis=0, ddof=1).mean() }[self.covariance_type] elif self.covariance_type in ['full', 'tied']: self.covariance_prior_ = check_array( self.covariance_prior, dtype=[np.float64, np.float32], ensure_2d=False) _check_shape(self.covariance_prior_, (n_features, n_features), '%s covariance_prior' % self.covariance_type) _check_precision_matrix(self.covariance_prior_, self.covariance_type) elif self.covariance_type == 'diag': self.covariance_prior_ = check_array( self.covariance_prior, dtype=[np.float64, np.float32], ensure_2d=False) _check_shape(self.covariance_prior_, (n_features,), '%s covariance_prior' % self.covariance_type) _check_precision_positivity(self.covariance_prior_, self.covariance_type) # spherical case elif self.covariance_prior > 0.: self.covariance_prior_ = self.covariance_prior else: raise ValueError("The parameter 'spherical covariance_prior' " "should be greater than 0., but got %.3f." % self.covariance_prior) def _initialize(self, X, resp): """Initialization of the mixture parameters. Parameters ---------- X : array-like, shape (n_samples, n_features) resp : array-like, shape (n_samples, n_components) """ nk, xk, sk = _estimate_gaussian_parameters(X, resp, self.reg_covar, self.covariance_type) self._estimate_weights(nk) self._estimate_means(nk, xk) self._estimate_precisions(nk, xk, sk) def _estimate_weights(self, nk): """Estimate the parameters of the Dirichlet distribution. Parameters ---------- nk : array-like, shape (n_components,) """ if self.weight_concentration_prior_type == 'dirichlet_process': # For dirichlet process weight_concentration will be a tuple # containing the two parameters of the beta distribution self.weight_concentration_ = ( 1. + nk, (self.weight_concentration_prior_ + np.hstack((np.cumsum(nk[::-1])[-2::-1], 0)))) else: # case Variationnal Gaussian mixture with dirichlet distribution self.weight_concentration_ = self.weight_concentration_prior_ + nk def _estimate_means(self, nk, xk): """Estimate the parameters of the Gaussian distribution. Parameters ---------- nk : array-like, shape (n_components,) xk : array-like, shape (n_components, n_features) """ self.mean_precision_ = self.mean_precision_prior_ + nk self.means_ = ((self.mean_precision_prior_ * self.mean_prior_ + nk[:, np.newaxis] * xk) / self.mean_precision_[:, np.newaxis]) def _estimate_precisions(self, nk, xk, sk): """Estimate the precisions parameters of the precision distribution. Parameters ---------- nk : array-like, shape (n_components,) xk : array-like, shape (n_components, n_features) sk : array-like The shape depends of `covariance_type`: 'full' : (n_components, n_features, n_features) 'tied' : (n_features, n_features) 'diag' : (n_components, n_features) 'spherical' : (n_components,) """ {"full": self._estimate_wishart_full, "tied": self._estimate_wishart_tied, "diag": self._estimate_wishart_diag, "spherical": self._estimate_wishart_spherical }[self.covariance_type](nk, xk, sk) self.precisions_cholesky_ = _compute_precision_cholesky( self.covariances_, self.covariance_type) def _estimate_wishart_full(self, nk, xk, sk): """Estimate the full Wishart distribution parameters. Parameters ---------- X : array-like, shape (n_samples, n_features) nk : array-like, shape (n_components,) xk : array-like, shape (n_components, n_features) sk : array-like, shape (n_components, n_features, n_features) """ _, n_features = xk.shape # Warning : in some Bishop book, there is a typo on the formula 10.63 # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk` is # the correct formula self.degrees_of_freedom_ = self.degrees_of_freedom_prior_ + nk self.covariances_ = np.empty((self.n_components, n_features, n_features)) for k in range(self.n_components): diff = xk[k] - self.mean_prior_ self.covariances_[k] = (self.covariance_prior_ + nk[k] * sk[k] + nk[k] * self.mean_precision_prior_ / self.mean_precision_[k] * np.outer(diff, diff)) # Contrary to the original bishop book, we normalize the covariances self.covariances_ /= ( self.degrees_of_freedom_[:, np.newaxis, np.newaxis]) def _estimate_wishart_tied(self, nk, xk, sk): """Estimate the tied Wishart distribution parameters. Parameters ---------- X : array-like, shape (n_samples, n_features) nk : array-like, shape (n_components,) xk : array-like, shape (n_components, n_features) sk : array-like, shape (n_features, n_features) """ _, n_features = xk.shape # Warning : in some Bishop book, there is a typo on the formula 10.63 # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk` # is the correct formula self.degrees_of_freedom_ = ( self.degrees_of_freedom_prior_ + nk.sum() / self.n_components) diff = xk - self.mean_prior_ self.covariances_ = ( self.covariance_prior_ + sk * nk.sum() / self.n_components + self.mean_precision_prior_ / self.n_components * np.dot( (nk / self.mean_precision_) * diff.T, diff)) # Contrary to the original bishop book, we normalize the covariances self.covariances_ /= self.degrees_of_freedom_ def _estimate_wishart_diag(self, nk, xk, sk): """Estimate the diag Wishart distribution parameters. Parameters ---------- X : array-like, shape (n_samples, n_features) nk : array-like, shape (n_components,) xk : array-like, shape (n_components, n_features) sk : array-like, shape (n_components, n_features) """ _, n_features = xk.shape # Warning : in some Bishop book, there is a typo on the formula 10.63 # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk` # is the correct formula self.degrees_of_freedom_ = self.degrees_of_freedom_prior_ + nk diff = xk - self.mean_prior_ self.covariances_ = ( self.covariance_prior_ + nk[:, np.newaxis] * ( sk + (self.mean_precision_prior_ / self.mean_precision_)[:, np.newaxis] * np.square(diff))) # Contrary to the original bishop book, we normalize the covariances self.covariances_ /= self.degrees_of_freedom_[:, np.newaxis] def _estimate_wishart_spherical(self, nk, xk, sk): """Estimate the spherical Wishart distribution parameters. Parameters ---------- X : array-like, shape (n_samples, n_features) nk : array-like, shape (n_components,) xk : array-like, shape (n_components, n_features) sk : array-like, shape (n_components,) """ _, n_features = xk.shape # Warning : in some Bishop book, there is a typo on the formula 10.63 # `degrees_of_freedom_k = degrees_of_freedom_0 + Nk` # is the correct formula self.degrees_of_freedom_ = self.degrees_of_freedom_prior_ + nk diff = xk - self.mean_prior_ self.covariances_ = ( self.covariance_prior_ + nk * ( sk + self.mean_precision_prior_ / self.mean_precision_ * np.mean(np.square(diff), 1))) # Contrary to the original bishop book, we normalize the covariances self.covariances_ /= self.degrees_of_freedom_ def _check_is_fitted(self): check_is_fitted(self, ['weight_concentration_', 'mean_precision_', 'means_', 'degrees_of_freedom_', 'covariances_', 'precisions_', 'precisions_cholesky_']) 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 nk, xk, sk = _estimate_gaussian_parameters( X, np.exp(log_resp), self.reg_covar, self.covariance_type) self._estimate_weights(nk) self._estimate_means(nk, xk) self._estimate_precisions(nk, xk, sk) def _estimate_log_weights(self): if self.weight_concentration_prior_type == 'dirichlet_process': digamma_sum = digamma(self.weight_concentration_[0] + self.weight_concentration_[1]) digamma_a = digamma(self.weight_concentration_[0]) digamma_b = digamma(self.weight_concentration_[1]) return (digamma_a - digamma_sum + np.hstack((0, np.cumsum(digamma_b - digamma_sum)[:-1]))) else: # case Variationnal Gaussian mixture with dirichlet distribution return (digamma(self.weight_concentration_) - digamma(np.sum(self.weight_concentration_))) def _estimate_log_prob(self, X): _, n_features = X.shape # We remove `n_features * np.log(self.degrees_of_freedom_)` because # the precision matrix is normalized log_gauss = (_estimate_log_gaussian_prob( X, self.means_, self.precisions_cholesky_, self.covariance_type) - .5 * n_features * np.log(self.degrees_of_freedom_)) log_lambda = n_features * np.log(2.) + np.sum(digamma( .5 * (self.degrees_of_freedom_ - np.arange(0, n_features)[:, np.newaxis])), 0) return log_gauss + .5 * (log_lambda - n_features / self.mean_precision_) def _compute_lower_bound(self, log_resp, log_prob_norm): """Estimate the lower bound of the model. The lower bound on the likelihood (of the training data with respect to the model) is used to detect the convergence and has to decrease at each iteration. Parameters ---------- X : array-like, shape (n_samples, n_features) log_resp : array, shape (n_samples, n_components) Logarithm of the posterior probabilities (or responsibilities) of the point of each sample in X. log_prob_norm : float Logarithm of the probability of each sample in X. Returns ------- lower_bound : float """ # Contrary to the original formula, we have done some simplification # and removed all the constant terms. n_features, = self.mean_prior_.shape # We removed `.5 * n_features * np.log(self.degrees_of_freedom_)` # because the precision matrix is normalized. log_det_precisions_chol = (_compute_log_det_cholesky( self.precisions_cholesky_, self.covariance_type, n_features) - .5 * n_features * np.log(self.degrees_of_freedom_)) if self.covariance_type == 'tied': log_wishart = self.n_components * np.float64(_log_wishart_norm( self.degrees_of_freedom_, log_det_precisions_chol, n_features)) else: log_wishart = np.sum(_log_wishart_norm( self.degrees_of_freedom_, log_det_precisions_chol, n_features)) if self.weight_concentration_prior_type == 'dirichlet_process': log_norm_weight = -np.sum(betaln(self.weight_concentration_[0], self.weight_concentration_[1])) else: log_norm_weight = _log_dirichlet_norm(self.weight_concentration_) return (-np.sum(np.exp(log_resp) * log_resp) - log_wishart - log_norm_weight - 0.5 * n_features * np.sum(np.log(self.mean_precision_))) def _get_parameters(self): return (self.weight_concentration_, self.mean_precision_, self.means_, self.degrees_of_freedom_, self.covariances_, self.precisions_cholesky_) def _set_parameters(self, params): (self.weight_concentration_, self.mean_precision_, self.means_, self.degrees_of_freedom_, self.covariances_, self.precisions_cholesky_) = params # Weights computation if self.weight_concentration_prior_type == "dirichlet_process": weight_dirichlet_sum = (self.weight_concentration_[0] + self.weight_concentration_[1]) tmp = self.weight_concentration_[1] / weight_dirichlet_sum self.weights_ = ( self.weight_concentration_[0] / weight_dirichlet_sum * np.hstack((1, np.cumprod(tmp[:-1])))) self.weights_ /= np.sum(self.weights_) else: self. weights_ = (self.weight_concentration_ / np.sum(self.weight_concentration_)) # Precisions matrices computation if self.covariance_type == 'full': self.precisions_ = np.array([ np.dot(prec_chol, prec_chol.T) for prec_chol in self.precisions_cholesky_]) elif self.covariance_type == 'tied': self.precisions_ = np.dot(self.precisions_cholesky_, self.precisions_cholesky_.T) else: self.precisions_ = self.precisions_cholesky_ ** 2