# Tutorial¶

hmmlearn implements the Hidden Markov Models (HMMs). The HMM is a generative probabilistic model, in which a sequence of observable $$\mathbf{X}$$ variables is generated by a sequence of internal hidden states $$\mathbf{Z}$$. The hidden states are not observed directly. The transitions between hidden states are assumed to have the form of a (first-order) Markov chain. They can be specified by the start probability vector $$\boldsymbol{\pi}$$ and a transition probability matrix $$\mathbf{A}$$. The emission probability of an observable can be any distribution with parameters $$\boldsymbol{\theta}$$ conditioned on the current hidden state. The HMM is completely determined by $$\boldsymbol{\pi}$$, $$\mathbf{A}$$ and $$\boldsymbol{\theta}$$.

There are three fundamental problems for HMMs:

• Given the model parameters and observed data, estimate the optimal sequence of hidden states.

• Given the model parameters and observed data, calculate the model likelihood.

• Given just the observed data, estimate the model parameters.

The first and the second problem can be solved by the dynamic programming algorithms known as the Viterbi algorithm and the Forward-Backward algorithm, respectively. The last one can be solved by an iterative Expectation-Maximization (EM) algorithm, known as the Baum-Welch algorithm.

References:

• Lawrence R. Rabiner “A tutorial on hidden Markov models and selected applications in speech recognition”, Proceedings of the IEEE 77.2, pp. 257-286, 1989.

• Jeff A. Bilmes, “A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models.”, 1998.

• Mark Stamp. “A revealing introduction to hidden Markov models”. Tech. rep. Department of Computer Science, San Jose State University, 2018. url: http://www.cs.sjsu.edu/~stamp/RUA/HMM.pdf.

## Available models¶

 hmm.GaussianHMM Hidden Markov Model with Gaussian emissions. hmm.GMMHMM Hidden Markov Model with Gaussian mixture emissions. hmm.MultinomialHMM Hidden Markov Model with multinomial (discrete) emissions.

Read on for details on how to implement a HMM with a custom emission probability.

## Building HMM and generating samples¶

You can build a HMM instance by passing the parameters described above to the constructor. Then, you can generate samples from the HMM by calling sample().

>>> import numpy as np
>>> from hmmlearn import hmm
>>> np.random.seed(42)
>>>
>>> model = hmm.GaussianHMM(n_components=3, covariance_type="full")
>>> model.startprob_ = np.array([0.6, 0.3, 0.1])
>>> model.transmat_ = np.array([[0.7, 0.2, 0.1],
...                             [0.3, 0.5, 0.2],
...                             [0.3, 0.3, 0.4]])
>>> model.means_ = np.array([[0.0, 0.0], [3.0, -3.0], [5.0, 10.0]])
>>> model.covars_ = np.tile(np.identity(2), (3, 1, 1))
>>> X, Z = model.sample(100)


The transition probability matrix need not to be ergodic. For instance, a left-right HMM can be defined as follows:

>>> lr = hmm.GaussianHMM(n_components=3, covariance_type="diag",
...                      init_params="cm", params="cmt")
>>> lr.startprob_ = np.array([1.0, 0.0, 0.0])
>>> lr.transmat_ = np.array([[0.5, 0.5, 0.0],
...                          [0.0, 0.5, 0.5],
...                          [0.0, 0.0, 1.0]])


If any of the required parameters are missing, sample() will raise an exception:

>>> model = hmm.GaussianHMM(n_components=3)
>>> X, Z = model.sample(100)
Traceback (most recent call last):
...
sklearn.exceptions.NotFittedError: This GaussianHMM instance is not fitted yet. Call 'fit' with appropriate arguments before using this method.


### Fixing parameters¶

Each HMM parameter has a character code which can be used to customize its initialization and estimation. The EM algorithm needs a starting point to proceed, thus prior to training each parameter is assigned a value either random or computed from the data. It is possible to hook into this process and provide a starting point explicitly. To do so

1. ensure that the character code for the parameter is missing from init_params and then

2. set the parameter to the desired value.

For example, consider a HMM with an explicitly initialized transition probability matrix:

>>> model = hmm.GaussianHMM(n_components=3, n_iter=100, init_params="mcs")
>>> model.transmat_ = np.array([[0.7, 0.2, 0.1],
...                             [0.3, 0.5, 0.2],
...                             [0.3, 0.3, 0.4]])


A similar trick applies to parameter estimation. If you want to fix some parameter at a specific value, remove the corresponding character from params and set the parameter value before training.

Examples:

## Training HMM parameters and inferring the hidden states¶

You can train an HMM by calling the fit() method. The input is a matrix of concatenated sequences of observations (aka samples) along with the lengths of the sequences (see Working with multiple sequences).

Note, since the EM algorithm is a gradient-based optimization method, it will generally get stuck in local optima. You should in general try to run fit with various initializations and select the highest scored model.

The score of the model can be calculated by the score() method.

The inferred optimal hidden states can be obtained by calling predict() method. The predict method can be specified with a decoder algorithm. Currently the Viterbi algorithm ("viterbi"), and maximum a posteriori estimation ("map") are supported.

This time, the input is a single sequence of observed values. Note, the states in remodel will have a different order than those in the generating model.

>>> remodel = hmm.GaussianHMM(n_components=3, covariance_type="full", n_iter=100)
>>> remodel.fit(X)
GaussianHMM(...
>>> Z2 = remodel.predict(X)


### Monitoring convergence¶

The number of EM algorithm iterations is upper bounded by the n_iter parameter. The training proceeds until n_iter steps were performed or the change in score is lower than the specified threshold tol. Note, that depending on the data, the EM algorithm may or may not achieve convergence in the given number of steps.

You can use the monitor_ attribute to diagnose convergence:

>>> remodel.monitor_
ConvergenceMonitor(
history=[...],
iter=10,
n_iter=100,
tol=0.01,
verbose=False,
)
>>> remodel.monitor_.converged
True


### Working with multiple sequences¶

All of the examples so far were using a single sequence of observations. The input format in the case of multiple sequences is a bit involved and is best understood by example.

Consider two 1D sequences:

>>> X1 = [[0.5], [1.0], [-1.0], [0.42], [0.24]]
>>> X2 = [[2.4], [4.2], [0.5], [-0.24]]


To pass both sequences to fit() or predict(), first concatenate them into a single array and then compute an array of sequence lengths:

>>> X = np.concatenate([X1, X2])
>>> lengths = [len(X1), len(X2)]


Finally, just call the desired method with X and lengths:

>>> hmm.GaussianHMM(n_components=3).fit(X, lengths)
GaussianHMM(...


After training, a HMM can be easily persisted for future use with the standard pickle module:

>>> import pickle
>>> with open("filename.pkl", "wb") as file: pickle.dump(remodel, file)
>>> with open("filename.pkl", "rb") as file: pickle.load(file)
GaussianHMM(...


## Implementing HMMs with custom emission probabilities¶

If you want to implement a custom emission probability (e.g. Poisson), you have to subclass BaseHMM and override the following methods

 Initialize model parameters prior to fitting. Validate model parameters prior to fitting. BaseHMM._generate_sample_from_state(state[, ...]) Generate a random sample from a given component. Compute per-component emission log probability under the model. Compute per-component probability under the model. Initialize sufficient statistics required for M-step. Update sufficient statistics from a given sample. BaseHMM._do_mstep(stats) Perform the M-step of EM algorithm.

Optionally, only one of _compute_likelihood and _compute_log_likelihood need to be overridden, and the base implementation will provide the other.