Gaussian Mixture HMM¶
The Gaussian Mixture HMM is a variant of HMM that uses a multivariate Gaussian mixture model as the emission distribution for each state.
This HMM variant can be used to recognize unbounded realvalued univariate or multivariate sequences.
Emissions¶
The emission distribution \(b_m\) of an observation \(\mathbf{o}^{(t)}\) at time \(t\) for state \(m\) is formed by a weighted mixture of \(K\) multivariate Gaussian probability density functions, defined as:
Where:
 \(\mathbf{o}^{(t)}=\left(o_1^{(t)}, o_2^{(t)}, \ldots, o_D^{(t)}\right)\) is a single observation at time \(t\), such that \(\mathbf{o}^{(t)}\in\mathbb{R}^D\).
 \(c_k^{(m)}\) is a component mixture weight for the \(k^\text{th}\) Gaussian component of the \(m^\text{th}\) state, such that \(\sum_{k=1}^K c_k^{(m)} = 1\) and \(c_k^{(m)}\in[0, 1]\).
 \(\boldsymbol\mu_k^{(m)}\) is a mean vector for the \(k^\text{th}\) Gaussian component of the \(m^\text{th}\) state, such that \(\boldsymbol\mu_k^{(m)}\in\mathbb{R}^D\).
 \(\Sigma_k^{(m)}\) is a covariance matrix for the \(k^\text{th}\) Gaussian component of the \(m^\text{th}\) state, such that \(\Sigma_k^{(m)}\in\mathbb{R}^{D\times D}\) and \(\Sigma_k^{(m)}\) is symmetric and positive semidefinite.
 \(\mathcal{N}_D\) is the \(D\)dimensional multivariate Gaussian probability density function.
Using a mixture rather than a single Gaussian allows for more flexible modelling of observations.
The component mixture weights, mean vector and covariance matrix for all states and Gaussian components are updated during training via ExpectationMaximization through the BaumWelch algorithm.
Covariance matrix types¶
The \(K\) covariance matrices for a state can come in different forms:
Full: All values are fully learnable independently for each component.
Diagonal: Only values along the diagonal may be learned independently for each component.
Spherical: Same as diagonal, with a single value shared along the diagonal for each component.
Tied: Same as full, with all components sharing the same single covariance matrix.
Estimating a full covariance matrix is not always necessary, particularly when a sufficient number of Gaussian components are used. If time is limiting, a spherical, diagonal and tied covariance matrix may also yield strong results while reducing training time due to having fewer parameters to estimate.
API reference¶
Class¶
A hidden Markov model with multivariate Gaussian mixture emissions. 
Methods¶

Initializes the 

The Akaike information criterion of the model, evaluated with the maximum likelihood of 

The Bayesian information criterion of the model, evaluated with the maximum likelihood of 

Fits the HMM to the sequences in 

Freezes the trainable parameters of the HMM, preventing them from being updated during the Baum—Welch algorithm. 

Retrieves the number of trainable parameters. 

Calculates the loglikelihood of the HMM generating a single observation sequence. 

Sets the initial state probabilities. 

Sets the covariance matrices of the state emission distributions. 

Sets the mean vectors of the state emission distributions. 

Sets the component mixture weights of the state emission distributions. 

Sets the transition probability matrix. 

Unfreezes the trainable parameters of the HMM, allowing them to be updated during the Baum—Welch algorithm. 
 class sequentia.models.hmm.variants.GaussianMixtureHMM[source]¶
A hidden Markov model with multivariate Gaussian mixture emissions.
Examples
Using a
GaussianMixtureHMM
to learn how to recognize spoken samples of the digit 3.See
load_digits()
for more information on the sample dataset used in this example.import numpy as np from sequentia.datasets import load_digits from sequentia.models.hmm import GaussianMixtureHMM # Seed for reproducible pseudorandomness random_state = np.random.RandomState(1) # Fetch MFCCs of spoken samples for the digit 3 data = load_digits(digits=[3]) train_data, test_data = data.split(test_size=0.2, random_state=random_state) # Create and train a GaussianMixtureHMM to recognize the digit 3 model = GaussianMixtureHMM(random_state=random_state) X_train, lengths_train = train_data.X_lengths model.fit(X_train, lengths_train) # Calculate the loglikelihood of the first test sample being generated by this model x, y = test_data[0] model.score(x)
 __init__(*, n_states=5, n_components=3, covariance_type='spherical', topology='leftright', random_state=None, hmmlearn_kwargs={'init_params': 'stmcw', 'params': 'stmcw'})[source]¶
Initializes the
GaussianMixtureHMM
. Parameters:
n_states (PositiveInt) – Number of states in the Markov chain.
n_components (PositiveInt) – Number of Gaussian components in the mixture emission distribution for each state.
covariance_type (Literal['spherical', 'diag', 'full', 'tied']) – Type of covariance matrix in the mixture emission distribution for each state  see Covariance matrix types.
topology (Literal['ergodic', 'leftright', 'linear']  None) –
Transition topology of the Markov chain — see Topologies.
If
None
, behaves the same as'ergodic'
but with hmmlearn initialization.
random_state (NonNegativeInt  RandomState  None) – Seed or
numpy.random.RandomState
object for reproducible pseudorandomness.hmmlearn_kwargs (Dict[str, Any]) – Additional keyword arguments provided to the hmmlearn HMM constructor.
 Return type:
 aic(X, lengths=None)[source]¶
The Akaike information criterion of the model, evaluated with the maximum likelihood of
X
. Parameters:
X (Array) –
Univariate or multivariate observation sequence(s).
Should be a single 1D or 2D array.
Should have length as the 1st dimension and features as the 2nd dimension.
Should be a concatenated sequence if multiple sequences are provided, with respective sequence lengths being provided in the
lengths
argument for decoding the original sequences.
lengths (Array  None) –
Lengths of the observation sequence(s) provided in
X
.If
None
, thenX
is assumed to be a single observation sequence.len(X)
should be equal tosum(lengths)
.
 Note:
This method requires a trained model — see
fit()
. Returns:
The Akaike information criterion.
 Return type:
float
 bic(X, lengths=None)[source]¶
The Bayesian information criterion of the model, evaluated with the maximum likelihood of
X
. Parameters:
X (Array) –
Univariate or multivariate observation sequence(s).
Should be a single 1D or 2D array.
Should have length as the 1st dimension and features as the 2nd dimension.
Should be a concatenated sequence if multiple sequences are provided, with respective sequence lengths being provided in the
lengths
argument for decoding the original sequences.
lengths (Array  None) –
Lengths of the observation sequence(s) provided in
X
.If
None
, thenX
is assumed to be a single observation sequence.len(X)
should be equal tosum(lengths)
.
 Note:
This method requires a trained model — see
fit()
. Returns:
The Bayesian information criterion.
 Return type:
float
 fit(X, lengths=None)[source]¶
Fits the HMM to the sequences in
X
, using the Baum—Welch algorithm. Parameters:
X (Array) –
Univariate or multivariate observation sequence(s).
Should be a single 1D or 2D array.
Should have length as the 1st dimension and features as the 2nd dimension.
Should be a concatenated sequence if multiple sequences are provided, with respective sequence lengths being provided in the
lengths
argument for decoding the original sequences.
lengths (Array  None) –
Lengths of the observation sequence(s) provided in
X
.If
None
, thenX
is assumed to be a single observation sequence.len(X)
should be equal tosum(lengths)
.
 Returns:
The fitted HMM.
 Return type:
 freeze(params='stmcw')[source]¶
Freezes the trainable parameters of the HMM, preventing them from being updated during the Baum—Welch algorithm.
 Parameters:
params (str) –
A string specifying which parameters to freeze. Can contain a combination of:
's'
for initial state probabilities,'t'
for transition probabilities,'m'
for emission distribution means,'c'
for emission distribution covariances,'w'
for emission distribution mixture weights.
 Note:
If used, this method should normally be called before
fit()
.
See also
unfreeze
Unfreezes the trainable parameters of the HMM, allowing them to be updated during the Baum—Welch algorithm.
 n_params()[source]¶
Retrieves the number of trainable parameters.
 Note:
This method requires a trained model — see
fit()
. Returns:
Number of trainable parameters.
 Return type:
NonNegativeInt
 score(x)[source]¶
Calculates the loglikelihood of the HMM generating a single observation sequence.
 Parameters:
x (Array) –
Univariate or multivariate observation sequence.
Should be a single 1D or 2D array.
Should have length as the 1st dimension and features as the 2nd dimension.
 Note:
This method requires a trained model — see
fit()
. Returns:
The loglikelihood.
 Return type:
float
 set_start_probs(values='random')¶
Sets the initial state probabilities.
If this method is not called, initial state probabilities are initialized depending on the value of
topology
provided to__init__()
.If
topology
was set to'ergodic'
,'leftright'
or'linear'
, then random probabilities will be assigned according to the topology by callingset_start_probs()
withvalue='random'
.If
topology
was set toNone
, then initial state probabilities will be initialized by hmmlearn.
 Parameters:
values (Array  Literal['uniform', 'random']) –
Probabilities or probability type to assign as initial state probabilities.
If an
Array
, should be a vector of starting probabilities for each state.If
'uniform'
, there is an equal probability of starting in any state.If
'random'
, the vector of initial state probabilities is sampled from a Dirichlet distribution with unit concentration parameters.
 Note:
If used, this method should normally be called before
fit()
.
 set_state_covariances(values)[source]¶
Sets the covariance matrices of the state emission distributions.
If this method is not called, covariance matrices will be initialized by hmmlearn.
 Parameters:
values (Array) – Array of emission distribution covariance values.
 Note:
If used, this method should normally be called before
fit()
.
 set_state_means(values)[source]¶
Sets the mean vectors of the state emission distributions.
If this method is not called, mean vectors will be initialized by hmmlearn.
 Parameters:
values (Array) – Array of emission distribution mean values.
 Note:
If used, this method should normally be called before
fit()
.
 set_state_weights(values)[source]¶
Sets the component mixture weights of the state emission distributions.
If this method is not called, component mixture weights will be initialized by hmmlearn.
 Parameters:
values (Array) – Array of emission distribution component mixture weights.
 Note:
If used, this method should normally be called before
fit()
.
 set_transitions(values='random')¶
Sets the transition probability matrix.
If this method is not called, transition probabilities are initialized depending on the value of
topology
provided to__init__()
:If
topology
was set to'ergodic'
,'leftright'
or'linear'
, then random probabilities will be assigned according to the topology by callingset_transitions()
withvalue='random'
.If
topology
was set toNone
, then initial state probabilities will be initialized by hmmlearn.
 Parameters:
values (Array  Literal['uniform', 'random']) –
Probabilities or probability type to assign as state transition probabilities.
If an
Array
, should be a matrix of probabilities where each row must some to one and represents the probabilities of transitioning out of a state.If
'uniform'
, for each state there is an equal probability of transitioning to any state permitted by the topology.If
'random'
, the vector of transition probabilities for each row is sampled from a Dirichlet distribution with unit concentration parameters, according to the shape of the topology.
 Note:
If used, this method should normally be called before
fit()
.
 unfreeze(params='stmcw')[source]¶
Unfreezes the trainable parameters of the HMM, allowing them to be updated during the Baum—Welch algorithm.
 Parameters:
params (str) –
A string specifying which parameters to unfreeze. Can contain a combination of:
's'
for initial state probabilities,'t'
for transition probabilities,'m'
for emission distribution means,'c'
for emission distribution covariances,'w'
for emission distribution mixture weights.
See also
freeze
Freezes the trainable parameters of the HMM, preventing them from be updated during the Baum—Welch algorithm.
 covariance_type¶
Type of covariance matrix in the emission model mixture distribution for each state.
 n_components¶
Number of Gaussian components in the emission model mixture distribution for each state.
 n_states¶
Number of states in the Markov chain.
 random_state¶
Seed or
numpy.random.RandomState
object for reproducible pseudorandomness.
 topology¶
Transition topology of the Markov chain — see Topologies.