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.. _hmm:
====================
Hidden Markov Models
====================
.. currentmodule:: sklearn.hmm
`sklearn.hmm` implements the algorithms of Hidden Markov Model (HMM).
HMM is a generative probabilistic model, in which a sequence of observable
:math:`\mathbf{X}` variable is generated by a sequence of internal hidden
state :math:`\mathbf{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 :math:`\boldsymbol{\Pi}` and the transition probability matrix
:math:`\mathbf{A}`.
The emission probability of observable can be any distribution with the
parameters :math:`\boldsymbol{{\Theta}_i}` conditioned on the current hidden
state index. (e.g. Multinomial, Gaussian).
Thus the HMM can be completely determined by
:math:`\boldsymbol{\Pi, \mathbf{A}}` and :math:`\boldsymbol{{\Theta}_i}`.
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.
.. topic:: References:
[Rabiner89] `A tutorial on hidden Markov models and selected applications in speech recognition <http://www.cs.ubc.ca/~murphyk/Bayes/rabiner.pdf>`_
Lawrence, R. Rabiner, 1989
Using HMM
=========
Classes in this module include :class:`MultinomalHMM` :class:`GaussianHMM`,
and :class:`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)
.. figure:: ../auto_examples/images/plot_hmm_sampling_1.png
:target: ../auto_examples/plot_hmm_sampling.html
:align: center
:scale: 75%
.. topic:: Examples:
* :ref:`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)
.. topic:: Examples:
* :ref:`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 :class:`_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`.
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