hmm.py

# Hidden Markov Models
#
# Author: Ron Weiss <ronweiss@gmail.com>
# and Shiqiao Du <lucidfrontier.45@gmail.com>

"""
The :mod:`sklearn.hmm` module implements hidden Markov models.

**Warning:** :mod:`sklearn.hmm` is orphaned, undocumented and has known
numerical stability issues. If nobody volunteers to write documentation and
make it more stable, this module will be removed in version 0.11.
"""

import string

import numpy as np

from .utils import check_random_state
from .utils.extmath import logsumexp
from .base import BaseEstimator
from .mixture import (
    GMM, log_multivariate_normal_density, sample_gaussian,
    distribute_covar_matrix_to_match_covariance_type, _validate_covars)
from . import cluster
from .utils import deprecated
from . import _hmmc


ZEROLOGPROB = -1e200
EPS = np.finfo(float).eps
NEGINF = -np.inf
decoder_algorithms = ("viterbi", "map")


def normalize(A, axis=None):
    """ Normalize the input array so that it sums to 1.

    Parameters
    ----------
    A: array, shape (n_samples, n_features)
       Non-normalized input data
    axis: int
          dimension along which normalization is performed

    Returns
    -------
    normalized_A: array, shape (n_samples, n_features)
        A with values normalized (summing to 1) along the prescribed axis

    WARNING: Modifies inplace the array
    """
    A += EPS
    Asum = A.sum(axis)
    if axis and A.ndim > 1:
        # Make sure we don't divide by zero.
        Asum[Asum == 0] = 1
        shape = list(A.shape)
        shape[axis] = 1
        Asum.shape = shape
    return A / Asum


class _BaseHMM(BaseEstimator):
    """Hidden Markov Model base class.

    Representation of a hidden Markov model probability distribution.
    This class allows for easy evaluation of, sampling from, and
    maximum-likelihood estimation of the parameters of a HMM.

    See the instance documentation for details specific to a
    particular object.

    Attributes
    ----------
    n_components : int
        Number of states in the model.

    transmat : array, shape (`n_components`, `n_components`)
        Matrix of transition probabilities between states.

    startprob : array, shape ('n_components`,)
        Initial state occupation distribution.

    transmat_prior : array, shape (`n_components`, `n_components`)
        Matrix of prior transition probabilities between states.

    startprob_prior : array, shape ('n_components`,)
        Initial state occupation prior distribution.

    algorithm : string, one of the decoder_algorithms
        decoder algorithm

    random_state: RandomState or an int seed (0 by default)
        A random number generator instance

    See Also
    --------
    GMM : Gaussian mixture model
    """

    # This class implements the public interface to all HMMs that
    # derive from it, including all of the machinery for the
    # forward-backward and Viterbi algorithms.  Subclasses need only
    # implement _generate_sample_from_state(), _compute_log_likelihood(),
    # _init(), _initialize_sufficient_statistics(),
    # _accumulate_sufficient_statistics(), and _do_mstep(), all of
    # which depend on the specific emission distribution.
    #
    # Subclasses will probably also want to implement properties for
    # the emission distribution parameters to expose them publically.

    def __init__(self, n_components=1, startprob=None, transmat=None,
            startprob_prior=None, transmat_prior=None,
            algorithm="viterbi", random_state=None):
        self.n_components = n_components

        if startprob is None:
            startprob = np.tile(1.0 / n_components, n_components)
        self.startprob_ = startprob

        if startprob_prior is None:
            startprob_prior = 1.0
        self.startprob_prior = startprob_prior

        if transmat is None:
            transmat = np.tile(1.0 / n_components,
                    (n_components, n_components))
        self.transmat_ = transmat

        if transmat_prior is None:
            transmat_prior = 1.0
        self.transmat_prior = transmat_prior

        if algorithm in decoder_algorithms:
            self._algorithm = algorithm
        else:
            self._algorithm = "viterbi"
        self.random_state = random_state

    def eval(self, obs):
        """Compute the log probability under the model and compute posteriors

        Implements rank and beam pruning in the forward-backward
        algorithm to speed up inference in large models.

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            Sequence of n_features-dimensional data points.  Each row
            corresponds to a single point in the sequence.

        Returns
        -------
        logprob : float
            Log likelihood of the sequence `obs`
        posteriors: array_like, shape (n, n_components)
            Posterior probabilities of each state for each
            observation

        See Also
        --------
        score : Compute the log probability under the model
        decode : Find most likely state sequence corresponding to a `obs`
        """
        obs = np.asarray(obs)
        framelogprob = self._compute_log_likelihood(obs)
        logprob, fwdlattice = self._do_forward_pass(framelogprob)
        bwdlattice = self._do_backward_pass(framelogprob)
        gamma = fwdlattice + bwdlattice
        # gamma is guaranteed to be correctly normalized by logprob at
        # all frames, unless we do approximate inference using pruning.
        # So, we will normalize each frame explicitly in case we
        # pruned too aggressively.
        posteriors = np.exp(gamma.T - logsumexp(gamma, axis=1)).T
        posteriors += np.finfo(np.float32).eps
        posteriors /= np.sum(posteriors, axis=1).reshape((-1, 1))
        return logprob, posteriors

    def score(self, obs):
        """Compute the log probability under the model.

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            Sequence of n_features-dimensional data points.  Each row
            corresponds to a single data point.

        Returns
        -------
        logprob : float
            Log likelihood of the `obs`

        See Also
        --------
        eval : Compute the log probability under the model and posteriors
        decode : Find most likely state sequence corresponding to a `obs`
        """
        obs = np.asarray(obs)
        framelogprob = self._compute_log_likelihood(obs)
        logprob, _ = self._do_forward_pass(framelogprob)
        return logprob

    def _decode_viterbi(self, obs):
        """Find most likely state sequence corresponding to `obs`.

        Uses the Viterbi algorithm.

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            List of n_features-dimensional data points.  Each row
            corresponds to a single data point.

        Returns
        -------
        viterbi_logprob : float
            Log probability of the maximum likelihood path through the HMM
        state_sequence : array_like, shape (n,)
            Index of the most likely states for each observation

        See Also
        --------
        eval : Compute the log probability under the model and posteriors
        score : Compute the log probability under the model
        """
        obs = np.asarray(obs)
        framelogprob = self._compute_log_likelihood(obs)
        viterbi_logprob, state_sequence = self._do_viterbi_pass(framelogprob)
        return viterbi_logprob, state_sequence

    def _decode_map(self, obs):
        """Find most likely state sequence corresponding to `obs`.

        Uses the maximum a posteriori estimation.

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            List of n_features-dimensional data points.  Each row
            corresponds to a single data point.

        Returns
        -------
        map_logprob : float
            Log probability of the maximum likelihood path through the HMM
        state_sequence : array_like, shape (n,)
            Index of the most likely states for each observation

        See Also
        --------
        eval : Compute the log probability under the model and posteriors
        score : Compute the log probability under the model
        """
        _, posteriors = self.eval(obs)
        state_sequence = np.argmax(posteriors, axis=1)
        map_logprob = np.max(posteriors, axis=1).sum()
        return map_logprob, state_sequence

    def decode(self, obs, algorithm="viterbi"):
        """Find most likely state sequence corresponding to `obs`.
        Uses the selected algorithm for decoding.

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            List of n_features-dimensional data points.  Each row
            corresponds to a single data point.

        algorithm : string, one of the `decoder_algorithms`
            decoder algorithm to be used

        Returns
        -------
        logprob : float
            Log probability of the maximum likelihood path through the HMM
        state_sequence : array_like, shape (n,)
            Index of the most likely states for each observation

        See Also
        --------
        eval : Compute the log probability under the model and posteriors
        score : Compute the log probability under the model
        """
        if self._algorithm in decoder_algorithms:
            algorithm = self._algorithm
        elif algorithm in decoder_algorithms:
            algorithm = algorithm
        decoder = {"viterbi": self._decode_viterbi,
                   "map": self._decode_map}
        logprob, state_sequence = decoder[algorithm](obs)
        return logprob, state_sequence

    def predict(self, obs, algorithm="viterbi"):
        """Find most likely state sequence corresponding to `obs`.

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            List of n_features-dimensional data points.  Each row
            corresponds to a single data point.

        Returns
        -------
        state_sequence : array_like, shape (n,)
            Index of the most likely states for each observation
        """
        _, state_sequence = self.decode(obs, algorithm)
        return state_sequence

    def predict_proba(self, obs):
        """Compute the posterior probability for each state in the model

        Parameters
        ----------
        obs : array_like, shape (n, n_features)
            List of n_features-dimensional data points.  Each row
            corresponds to a single data point.

        Returns
        -------
        T : array-like, shape (n, n_components)
            Returns the probability of the sample for each state in the model.
        """
        _, posteriors = self.eval(obs)
        return posteriors

    def sample(self, n=1, random_state=None):
        """Generate random samples from the model.

        Parameters
        ----------
        n : int
            Number of samples to generate.

        random_state: RandomState or an int seed (0 by default)
            A random number generator instance. If None is given, the
            object's random_state is used

        Returns
        -------
        (obs, hidden_states)
        obs : array_like, length `n` List of samples
        hidden_states : array_like, length `n` List of hidden states
        """
        if random_state is None:
            random_state = self.random_state
        random_state = check_random_state(random_state)

        startprob_pdf = self.startprob_
        startprob_cdf = np.cumsum(startprob_pdf)
        transmat_pdf = self.transmat_
        transmat_cdf = np.cumsum(transmat_pdf, 1)

        # Initial state.
        rand = random_state.rand()
        currstate = (startprob_cdf > rand).argmax()
        hidden_states = [currstate]
        obs = [self._generate_sample_from_state(
                                currstate, random_state=random_state)]

        for _ in xrange(n - 1):
            rand = random_state.rand()
            currstate = (transmat_cdf[currstate] > rand).argmax()
            hidden_states.append(currstate)
            obs.append(self._generate_sample_from_state(
                                currstate, random_state=random_state))

        return np.array(obs), np.array(hidden_states, dtype=int)

    @deprecated("rvs is deprecated in 0.11 will be removed in 0.13:"
            + " use sample instead")
    def rvs(self, n=1, random_state=None):
        return self.sample(n, random_state)

    def fit(self, obs, n_iter=10, thresh=1e-2, params=string.ascii_letters,
            init_params=string.ascii_letters, **kwargs):
        """Estimate model parameters.

        An 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 ''. Likewise, if you
        would like just to do an initialization, call this method with
        n_iter=0.

        Parameters
        ----------
        obs : list
            List of array-like observation sequences (shape (n_i, n_features)).

        n_iter : int, optional
            Number of iterations to perform.

        thresh : float, optional
            Convergence threshold.

        params : string, optional
            Controls which parameters are updated in the training
            process.  Can contain any combination of 's' for startprob,
            't' for transmat, 'm' for means, and 'c' for covars, etc.
            Defaults to all parameters.

        init_params : string, optional
            Controls which parameters are initialized prior to
            training.  Can contain any combination of 's' for
            startprob, 't' for transmat, 'm' for means, and 'c' for
            covars, etc.  Defaults to all parameters.

        Notes
        -----
        In general, `logprob` should be non-decreasing unless
        aggressive pruning is used.  Decreasing `logprob` is generally
        a sign of overfitting (e.g. a covariance parameter getting too
        small).  You can fix this by getting more training data, or
        decreasing `covars_prior`.
        """
        self._init(obs, init_params)

        logprob = []
        for i in xrange(n_iter):
            # Expectation step
            stats = self._initialize_sufficient_statistics()
            curr_logprob = 0
            for seq in obs:
                framelogprob = self._compute_log_likelihood(seq)
                lpr, fwdlattice = self._do_forward_pass(framelogprob)
                bwdlattice = self._do_backward_pass(framelogprob)
                gamma = fwdlattice + bwdlattice
                posteriors = np.exp(gamma.T - logsumexp(gamma, axis=1)).T
                curr_logprob += lpr
                self._accumulate_sufficient_statistics(
                    stats, seq, framelogprob, posteriors, fwdlattice,
                    bwdlattice, params)
            logprob.append(curr_logprob)

            # Check for convergence.
            if i > 0 and abs(logprob[-1] - logprob[-2]) < thresh:
                break

            # Maximization step
            self._do_mstep(stats, params)

        return self

    def _get_algorithm(self):
        "decoder algorithm"
        return self._algorithm

    def _set_algorithm(self, algorithm):
        if algorithm not in decoder_algorithms:
            raise ValueError("algorithm must be one of the decoder_algorithms")
        self._algorithm = algorithm

    algorithm = property(_get_algorithm, _set_algorithm)

    def _get_startprob(self):
        """Mixing startprob for each state."""
        return np.exp(self._log_startprob)

    def _set_startprob(self, startprob):
        if len(startprob) != self.n_components:
            raise ValueError('startprob must have length n_components')
        if not np.allclose(np.sum(startprob), 1.0):
            raise ValueError('startprob must sum to 1.0')

        self._log_startprob = np.log(np.asarray(startprob).copy())

    startprob_ = property(_get_startprob, _set_startprob)

    def _get_transmat(self):
        """Matrix of transition probabilities."""
        return np.exp(self._log_transmat)

    def _set_transmat(self, transmat):
        if (np.asarray(transmat).shape
                != (self.n_components, self.n_components)):
            raise ValueError('transmat must have shape ' +
                    '(n_components, n_components)')
        if not np.all(np.allclose(np.sum(transmat, axis=1), 1.0)):
            raise ValueError('Rows of transmat must sum to 1.0')

        self._log_transmat = np.log(np.asarray(transmat).copy())
        underflow_idx = np.isnan(self._log_transmat)
        self._log_transmat[underflow_idx] = NEGINF

    transmat_ = property(_get_transmat, _set_transmat)

    def _do_viterbi_pass(self, framelogprob):
        n_observations, n_components = framelogprob.shape
        state_sequence, logprob = _hmmc._viterbi(
                n_observations, n_components, self._log_startprob,
                self._log_transmat, framelogprob)
        return logprob, state_sequence

    def _do_forward_pass(self, framelogprob):

        n_observations, n_components = framelogprob.shape
        fwdlattice = np.zeros((n_observations, n_components))
        _hmmc._forward(n_observations, n_components, self._log_startprob,
                self._log_transmat, framelogprob, fwdlattice)
        fwdlattice[fwdlattice <= ZEROLOGPROB] = NEGINF
        return logsumexp(fwdlattice[-1]), fwdlattice

    def _do_backward_pass(self, framelogprob):
        n_observations, n_components = framelogprob.shape
        bwdlattice = np.zeros((n_observations, n_components))
        _hmmc._backward(n_observations, n_components,
                self._log_startprob, self._log_transmat,
                framelogprob, bwdlattice)

        bwdlattice[bwdlattice <= ZEROLOGPROB] = NEGINF

        return bwdlattice

    def _compute_log_likelihood(self, obs):
        pass

    def _generate_sample_from_state(self, state, random_state=None):
        pass

    def _init(self, obs, params):
        if 's' in params:
            self.startprob_[:] = 1.0 / self.n_components
        if 't' in params:
            self.transmat_[:] = 1.0 / self.n_components

    # Methods used by self.fit()

    def _initialize_sufficient_statistics(self):
        stats = {'nobs': 0,
                 'start': np.zeros(self.n_components),
                 'trans': np.zeros((self.n_components, self.n_components))}
        return stats

    def _accumulate_sufficient_statistics(self, stats, seq, framelogprob,
                                          posteriors, fwdlattice, bwdlattice,
                                          params):
        stats['nobs'] += 1
        if 's' in params:
            stats['start'] += posteriors[0]
        if 't' in params:
            if _hmmc:
                n_observations, n_components = framelogprob.shape
                lneta = np.zeros((n_observations - 1,
                            n_components, n_components))
                lnP = logsumexp(fwdlattice[-1])
                _hmmc._compute_lneta(n_observations, n_components,
                        fwdlattice, self._log_transmat, bwdlattice,
                        framelogprob, lnP, lneta)
                stats["trans"] += np.exp(logsumexp(lneta, 0))
            else:
                for t in xrange(len(framelogprob)):
                    zeta = (fwdlattice[t - 1][:, np.newaxis]
                            + self._log_transmat + framelogprob[t]
                            + bwdlattice[t])
                    stats['trans'] += np.exp(zeta - logsumexp(zeta))

    def _do_mstep(self, stats, params):
        # Based on Huang, Acero, Hon, "Spoken Language Processing",
        # p. 443 - 445
        if 's' in params:
            self.startprob_ = normalize(
                np.maximum(self.startprob_prior - 1.0 + stats['start'], 1e-20))
        if 't' in params:
            self.transmat_ = normalize(
                np.maximum(self.transmat_prior - 1.0 + stats['trans'], 1e-20),
                axis=1)


class GaussianHMM(_BaseHMM):
    """Hidden Markov Model with Gaussian emissions

    Representation of a hidden Markov model probability distribution.
    This class allows for easy evaluation of, sampling from, and
    maximum-likelihood estimation of the parameters of a HMM.

    Parameters
    ----------
    n_components : int
        Number of states.

    _covariance_type : string
        String describing the type of covariance parameters to
        use.  Must be one of 'spherical', 'tied', 'diag', 'full'.
        Defaults to 'diag'.

    Attributes
    ----------
    covariance_type : string
        String describing the type of covariance parameters used by
        the model.  Must be one of 'spherical', 'tied', 'diag', 'full'.

    n_features : int
        Dimensionality of the Gaussian emissions.

    n_components : int
        Number of states in the model.

    transmat : array, shape (`n_components`, `n_components`)
        Matrix of transition probabilities between states.

    startprob : array, shape ('n_components`,)
        Initial state occupation distribution.

    means : array, shape (`n_components`, `n_features`)
        Mean parameters for each state.

    covars : array
        Covariance parameters for each state.  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: RandomState or an int seed (0 by default)
        A random number generator instance

    Examples
    --------
    >>> from sklearn.hmm import GaussianHMM
    >>> GaussianHMM(n_components=2)
    ...                             #doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
    GaussianHMM(algorithm='viterbi', covariance_type='diag', covars_prior=0.01,
        covars_weight=1, means_prior=None, means_weight=0, n_components=2,
        random_state=None, startprob=None, startprob_prior=1.0, transmat=None,
        transmat_prior=1.0)

    See Also
    --------
    GMM : Gaussian mixture model
    """

    def __init__(self, n_components=1, covariance_type='diag', startprob=None,
                 transmat=None, startprob_prior=None, transmat_prior=None,
                 algorithm="viterbi", means_prior=None, means_weight=0,
                 covars_prior=1e-2, covars_weight=1,
                 random_state=None):
        _BaseHMM.__init__(self, n_components, startprob, transmat,
                                        startprob_prior=startprob_prior,
                                        transmat_prior=transmat_prior,
                                        algorithm=algorithm,
                                        random_state=random_state)

        self._covariance_type = covariance_type
        if not covariance_type in ['spherical', 'tied', 'diag', 'full']:
            raise ValueError('bad covariance_type')

        self.means_prior = means_prior
        self.means_weight = means_weight

        self.covars_prior = covars_prior
        self.covars_weight = covars_weight

    # Read-only properties.
    @property
    def covariance_type(self):
        """Covariance type of the model.

        Must be one of 'spherical', 'tied', 'diag', 'full'.
        """
        return self._covariance_type

    def _get_means(self):
        """Mean parameters for each state."""
        return self._means_

    def _set_means(self, means):
        means = np.asarray(means)
        if hasattr(self, 'n_features') and \
               means.shape != (self.n_components, self.n_features):
            raise ValueError('means must have shape' +
                    '(n_components, n_features)')
        self._means_ = means.copy()
        self.n_features = self._means_.shape[1]

    means_ = property(_get_means, _set_means)

    def _get_covars(self):
        """Return covars as a full matrix."""
        if self._covariance_type == 'full':
            return self._covars_
        elif self._covariance_type == 'diag':
            return [np.diag(cov) for cov in self._covars_]
        elif self._covariance_type == 'tied':
            return [self._covars_] * self.n_components
        elif self._covariance_type == 'spherical':
            return [np.eye(self.n_features) * f for f in self._covars_]

    def _set_covars(self, covars):
        covars = np.asarray(covars)
        _validate_covars(covars, self._covariance_type, self.n_components)
        self._covars_ = covars.copy()

    covars_ = property(_get_covars, _set_covars)

    def _compute_log_likelihood(self, obs):
        return log_multivariate_normal_density(
            obs, self._means_, self._covars_, self._covariance_type)

    def _generate_sample_from_state(self, state, random_state=None):
        if self._covariance_type == 'tied':
            cv = self._covars_
        else:
            cv = self._covars_[state]
        return sample_gaussian(self._means_[state], cv, self._covariance_type,
                               random_state=random_state)

    def _init(self, obs, params='stmc'):
        super(GaussianHMM, self)._init(obs, params=params)

        if (hasattr(self, 'n_features')
            and self.n_features != obs[0].shape[1]):
            raise ValueError('Unexpected number of dimensions, got %s but '
                             'expected %s' % (obs[0].shape[1],
                                              self.n_features))

        self.n_features = obs[0].shape[1]

        if 'm' in params:
            self._means_ = cluster.KMeans(
                k=self.n_components).fit(obs[0]).cluster_centers_
        if 'c' in params:
            cv = np.cov(obs[0].T)
            if not cv.shape:
                cv.shape = (1, 1)
            self._covars_ = distribute_covar_matrix_to_match_covariance_type(
                cv, self._covariance_type, self.n_components)

    def _initialize_sufficient_statistics(self):
        stats = super(GaussianHMM, self)._initialize_sufficient_statistics()
        stats['post'] = np.zeros(self.n_components)
        stats['obs'] = np.zeros((self.n_components, self.n_features))
        stats['obs**2'] = np.zeros((self.n_components, self.n_features))
        stats['obs*obs.T'] = np.zeros((self.n_components, self.n_features,
                                       self.n_features))
        return stats

    def _accumulate_sufficient_statistics(self, stats, obs, framelogprob,
                                          posteriors, fwdlattice, bwdlattice,
                                          params):
        super(GaussianHMM, self)._accumulate_sufficient_statistics(
            stats, obs, framelogprob, posteriors, fwdlattice, bwdlattice,
            params)

        if 'm' in params or 'c' in params:
            stats['post'] += posteriors.sum(axis=0)
            stats['obs'] += np.dot(posteriors.T, obs)

        if 'c' in params:
            if self._covariance_type in ('spherical', 'diag'):
                stats['obs**2'] += np.dot(posteriors.T, obs ** 2)
            elif self._covariance_type in ('tied', 'full'):
                for t, o in enumerate(obs):
                    obsobsT = np.outer(o, o)
                    for c in xrange(self.n_components):
                        stats['obs*obs.T'][c] += posteriors[t, c] * obsobsT

    def _do_mstep(self, stats, params):
        super(GaussianHMM, self)._do_mstep(stats, params)

        # Based on Huang, Acero, Hon, "Spoken Language Processing",
        # p. 443 - 445
        denom = stats['post'][:, np.newaxis]
        if 'm' in params:
            prior = self.means_prior
            weight = self.means_weight
            if prior is None:
                weight = 0
                prior = 0
            self._means_ = (weight * prior + stats['obs']) / (weight + denom)

        if 'c' in params:
            covars_prior = self.covars_prior
            covars_weight = self.covars_weight
            if covars_prior is None:
                covars_weight = 0
                covars_prior = 0

            means_prior = self.means_prior
            means_weight = self.means_weight
            if means_prior is None:
                means_weight = 0
                means_prior = 0
            meandiff = self._means_ - means_prior

            if self._covariance_type in ('spherical', 'diag'):
                cv_num = (means_weight * (meandiff) ** 2
                          + stats['obs**2']
                          - 2 * self._means_ * stats['obs']
                          + self._means_ ** 2 * denom)
                cv_den = max(covars_weight - 1, 0) + denom
                self._covars_ = (covars_prior + cv_num) / cv_den
                if self._covariance_type == 'spherical':
                    self._covars_ = np.tile(self._covars_.mean(1)
                            [:, np.newaxis], (1, self._covars_.shape[1]))
            elif self._covariance_type in ('tied', 'full'):
                cvnum = np.empty((self.n_components, self.n_features,
                                  self.n_features))
                for c in xrange(self.n_components):
                    obsmean = np.outer(stats['obs'][c], self._means_[c])

                    cvnum[c] = (means_weight * np.outer(meandiff[c],
                                                        meandiff[c])
                                + stats['obs*obs.T'][c]
                                - obsmean - obsmean.T
                                + np.outer(self._means_[c], self._means_[c])
                                * stats['post'][c])
                cvweight = max(covars_weight - self.n_features, 0)
                if self._covariance_type == 'tied':
                    self._covars_ = ((covars_prior + cvnum.sum(axis=0))
                                    / (cvweight + stats['post'].sum()))
                elif self._covariance_type == 'full':
                    self._covars_ = ((covars_prior + cvnum)
                                   / (cvweight + stats['post'][:, None, None]))


class MultinomialHMM(_BaseHMM):
    """Hidden Markov Model with multinomial (discrete) emissions

    Attributes
    ----------
    n_components : int
        Number of states in the model.

    n_symbols : int
        Number of possible symbols emitted by the model (in the observations).

    transmat : array, shape (`n_components`, `n_components`)
        Matrix of transition probabilities between states.

    startprob : array, shape ('n_components`,)
        Initial state occupation distribution.

    emissionprob : array, shape ('n_components`, 'n_symbols`)
        Probability of emitting a given symbol when in each state.

    random_state: RandomState or an int seed (0 by default)
        A random number generator instance

    Examples
    --------
    >>> from sklearn.hmm import MultinomialHMM
    >>> MultinomialHMM(n_components=2)
    ...                             #doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
    MultinomialHMM(algorithm='viterbi', n_components=2, random_state=None,
                   startprob=None, startprob_prior=1.0, transmat=None,
                   transmat_prior=1.0)

    See Also
    --------
    GaussianHMM : HMM with Gaussian emissions
    """

    def __init__(self, n_components=1, startprob=None, transmat=None,
            startprob_prior=None, transmat_prior=None,
            algorithm="viterbi", random_state=None):
        """Create a hidden Markov model with multinomial emissions.

        Parameters
        ----------
        n_components : int
            Number of states.
        """
        _BaseHMM.__init__(self, n_components, startprob, transmat,
                                             startprob_prior=startprob_prior,
                                             transmat_prior=transmat_prior,
                                             algorithm=algorithm,
                                             random_state=random_state)

    def _get_emissionprob(self):
        """Emission probability distribution for each state."""
        return np.exp(self._log_emissionprob)

    def _set_emissionprob(self, emissionprob):
        emissionprob = np.asarray(emissionprob)
        if hasattr(self, 'n_symbols') and \
               emissionprob.shape != (self.n_components, self.n_symbols):
            raise ValueError('emissionprob must have shape '
                             '(n_components, n_symbols)')

        self._log_emissionprob = np.log(emissionprob)
        underflow_idx = np.isnan(self._log_emissionprob)
        self._log_emissionprob[underflow_idx] = NEGINF
        self.n_symbols = self._log_emissionprob.shape[1]

    emissionprob_ = property(_get_emissionprob, _set_emissionprob)

    def _compute_log_likelihood(self, obs):
        return self._log_emissionprob[:, obs].T

    def _generate_sample_from_state(self, state, random_state=None):
        cdf = np.cumsum(self.emissionprob_[state, :])
        random_state = check_random_state(random_state)
        rand = random_state.rand()
        symbol = (cdf > rand).argmax()
        return symbol

    def _init(self, obs, params='ste'):
        super(MultinomialHMM, self)._init(obs, params=params)

        if 'e' in params:
            emissionprob = normalize(np.random.rand(self.n_components,
                                                    self.n_symbols), 1)
            self.emissionprob_ = emissionprob

    def _initialize_sufficient_statistics(self):
        stats = super(MultinomialHMM, self)._initialize_sufficient_statistics()
        stats['obs'] = np.zeros((self.n_components, self.n_symbols))
        return stats

    def _accumulate_sufficient_statistics(self, stats, obs, framelogprob,
                                          posteriors, fwdlattice, bwdlattice,
                                          params):
        super(MultinomialHMM, self)._accumulate_sufficient_statistics(
            stats, obs, framelogprob, posteriors, fwdlattice, bwdlattice,
            params)
        if 'e' in params:
            for t, symbol in enumerate(obs):
                stats['obs'][:, symbol] += posteriors[t]

    def _do_mstep(self, stats, params):
        super(MultinomialHMM, self)._do_mstep(stats, params)
        if 'e' in params:
            self.emissionprob_ = (stats['obs']
                                 / stats['obs'].sum(1)[:, np.newaxis])


class GMMHMM(_BaseHMM):
    """Hidden Markov Model with Gaussin mixture emissions

    Attributes
    ----------
    n_components : int
        Number of states in the model.

    transmat : array, shape (`n_components`, `n_components`)
        Matrix of transition probabilities between states.

    startprob : array, shape ('n_components`,)
        Initial state occupation distribution.

    gmms : array of GMM objects, length `n_components`
        GMM emission distributions for each state.

    random_state: RandomState or an int seed (0 by default)
        A random number generator instance

    Examples
    --------
    >>> from sklearn.hmm import GMMHMM
    >>> GMMHMM(n_components=2, n_mix=10, covariance_type='diag')
    ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    GMMHMM(algorithm='viterbi', covariance_type='diag', covars_prior=0.01,
        gmms=[GMM(covariance_type=None, min_covar=0.001, n_components=10,
        random_state=None, thresh=0.01), GMM(covariance_type=None,
        min_covar=0.001, n_components=10, random_state=None, thresh=0.01)],
        n_components=2, n_mix=10, random_state=None, startprob=None,
        startprob_prior=1.0, transmat=None, transmat_prior=1.0)

    See Also
    --------
    GaussianHMM : HMM with Gaussian emissions
    """

    def __init__(self, n_components=1, n_mix=1, startprob=None, transmat=None,
            startprob_prior=None, transmat_prior=None, algorithm="viterbi",
            gmms=None, covariance_type='diag', covars_prior=1e-2,
            random_state=None):
        """Create a hidden Markov model with GMM emissions.

        Parameters
        ----------
        n_components : int
            Number of states.
        """
        _BaseHMM.__init__(self, n_components, startprob, transmat,
                                     startprob_prior=startprob_prior,
                                     transmat_prior=transmat_prior,
                                     algorithm=algorithm,
                                     random_state=random_state)

        # XXX: Hotfit for n_mix that is incompatible with the scikit's
        # BaseEstimator API
        self.n_mix = n_mix
        self._covariance_type = covariance_type
        self.covars_prior = covars_prior
        if gmms is None:
            gmms = []
            for x in xrange(self.n_components):
                if covariance_type is None:
                    g = GMM(n_mix)
                else:
                    g = GMM(n_mix, covariance_type=covariance_type)
                gmms.append(g)
        self.gmms = gmms

    # Read-only properties.
    @property
    def covariance_type(self):
        """Covariance type of the model.

        Must be one of 'spherical', 'tied', 'diag', 'full'.
        """
        return self._covariance_type

    def _compute_log_likelihood(self, obs):
        return np.array([g.score(obs) for g in self.gmms]).T

    def _generate_sample_from_state(self, state, random_state=None):
        return self.gmms[state].sample(1, random_state=random_state).flatten()

    def _init(self, obs, params='stwmc'):
        super(GMMHMM, self)._init(obs, params=params)

        allobs = np.concatenate(obs, 0)
        for g in self.gmms:
            g.fit(allobs, n_iter=0, init_params=params)

    def _initialize_sufficient_statistics(self):
        stats = super(GMMHMM, self)._initialize_sufficient_statistics()
        stats['norm'] = [np.zeros(g.weights_.shape) for g in self.gmms]
        stats['means'] = [np.zeros(np.shape(g.means_)) for g in self.gmms]
        stats['covars'] = [np.zeros(np.shape(g.covars_)) for g in self.gmms]
        return stats

    def _accumulate_sufficient_statistics(self, stats, obs, framelogprob,
                                          posteriors, fwdlattice, bwdlattice,
                                          params):
        super(GMMHMM, self)._accumulate_sufficient_statistics(
            stats, obs, framelogprob, posteriors, fwdlattice, bwdlattice,
            params)

        for state, g in enumerate(self.gmms):
            _, lgmm_posteriors = g.eval(obs)
            lgmm_posteriors += np.log(posteriors[:, state][:, np.newaxis]
                                      + np.finfo(np.float).eps)
            gmm_posteriors = np.exp(lgmm_posteriors)
            tmp_gmm = GMM(g.n_components, covariance_type=g._covariance_type)
            n_features = g.means_.shape[1]
            tmp_gmm._set_covars(
                distribute_covar_matrix_to_match_covariance_type(
                    np.eye(n_features), g._covariance_type,
                    g.n_components))
            norm = tmp_gmm._do_mstep(obs, gmm_posteriors, params)

            if np.any(np.isnan(tmp_gmm.covars_)):
                raise ValueError

            stats['norm'][state] += norm
            if 'm' in params:
                stats['means'][state] += tmp_gmm.means_ * norm[:, np.newaxis]
            if 'c' in params:
                if tmp_gmm._covariance_type == 'tied':
                    stats['covars'][state] += tmp_gmm.covars_ * norm.sum()
                else:
                    cvnorm = np.copy(norm)
                    shape = np.ones(tmp_gmm.covars_.ndim)
                    shape[0] = np.shape(tmp_gmm.covars_)[0]
                    cvnorm.shape = shape
                    stats['covars'][state] += tmp_gmm.covars_ * cvnorm

    def _do_mstep(self, stats, params):
        super(GMMHMM, self)._do_mstep(stats, params)
        # All that is left to do is to apply covars_prior to the
        # parameters updated in _accumulate_sufficient_statistics.
        for state, g in enumerate(self.gmms):
            n_features = g.means_.shape[1]
            norm = stats['norm'][state]
            if 'w' in params:
                g.weights_ = normalize(norm)
            if 'm' in params:
                g.means_ = stats['means'][state] / norm[:, np.newaxis]
            if 'c' in params:
                if g._covariance_type == 'tied':
                    g.covars_ = ((stats['covars'][state]
                                 + self.covars_prior * np.eye(n_features))
                                / norm.sum())
                else:
                    cvnorm = np.copy(norm)
                    shape = np.ones(g.covars_.ndim)
                    shape[0] = np.shape(g.covars_)[0]
                    cvnorm.shape = shape
                    if (g._covariance_type in ['spherical', 'diag']):
                        g.covars_ = (stats['covars'][state]
                                    + self.covars_prior) / cvnorm
                    elif g._covariance_type == 'full':
                        eye = np.eye(n_features)
                        g.covars_ = ((stats['covars'][state]
                                     + self.covars_prior * eye[np.newaxis])
                                    / cvnorm)