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Hidden Markov Models

sklearn.hmm

sklearn.hmm implements the algorithms of Hidden Markov Model (HMM). HMM is a generative probabilistic model, in which a sequence of observable X variable is generated by a sequence of internal hidden state Z. The hidden states can not be observed directly. The transition of hidden states is aussumed to be the first order Markov Chain. It can be specified by the start probability vector Π and the transition probability matrix A. The emission probability of observable can be any distribution with the parameters Θi conditioned on the current hidden state index. (e.g. Multinomial, Gaussian). Thus the HMM can be completely determined by Π, A and Θi.

There are three fundamental problems of HMM:

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

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

See the ref listed below for further detailed information.

Using HMM

Classes in this module include MultinomalHMM GaussianHMM, and GMMHMM. They implement HMM with emission probability of Multimomial distribution, Gaussian distribution and the mixture of Gaussian distributions.

Building HMM and generating samples

You can build an 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 sklearn import hmm

>>> startprob = np.array([0.6, 0.3, 0.1])
>>> transmat = np.array([[0.7, 0.2, 0.1], [0.3, 0.5, 0.2], [0.3, 0.3, 0.4]])
>>> means = np.array([[0.0, 0.0], [3.0, -3.0], [5.0, 10.0]])
>>> covars = np.tile(np.identity(2), (3, 1, 1))
>>> model = hmm.GaussianHMM(3, "full", startprob, transmat)
>>> model.means_ = means
>>> model.covars_ = covars
>>> X, Z = model.sample(100)

Examples:

  • example_plot_hmm_sampling.py

Training HMM parameters and infering the hidden states

You can train the HMM by calling fit method. The input is "the list" of the sequence of observed value. Note, since EM-algorithm is a gradient based optimization method, it will generally be stuck at local optimal. You should try to run fit with various initialization and select the highest scored model. The score of the model can be calculated by the score method. The infered optimal hidden states can be obtained by calling predict method. The predict method can be specified with decoder algorithm. Currently Viterbi algorithm viterbi, and maximum a posteriori estimation map is supported. This time, the input is a single sequence of observed values.:

>>> model2 = hmm.GaussianHMM(3, "full")
>>> model2.fit([X])
GaussianHMM(algorithm='viterbi', covariance_type='full', covars_prior=0.01,
      covars_weight=1, means_prior=None, means_weight=0, n_components=3,
      random_state=None, startprob=None, startprob_prior=1.0,
      transmat=None, transmat_prior=1.0)
>>> Z2 = model.predict(X)

Examples:

  • example_plot_hmm_stock_analysis.py

Implementing HMMs with other emission probabilities

If you want to implement other emission probability (e.g. Poisson), you have to make you own HMM class by inheriting the _BaseHMM and override necessary methods. They should be __init__, _compute_log_likelihood, _set and _get for addiitional parameters, _initialize_sufficient_statistics, _accumulate_sufficient_statistics and _do_mstep.