.. _hmm:

====================
Hidden Markov Models
====================

.. currentmodule:: sklearn.hmm

`sklearn.hmm` implements the Hidden Markov Models (HMMs).
The 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 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 :math:`\boldsymbol{\Pi}` and a transition probability matrix 
:math:`\mathbf{A}`.
The emission probability of an observable can be any distribution with
parameters :math:`\boldsymbol{{\Theta}_i}`
conditioned on the current hidden state (e.g. multinomial, Gaussian).
The HMM is completely determined by 
:math:`\boldsymbol{\Pi, \mathbf{A}}` and :math:`\boldsymbol{{\Theta}_i}`.

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 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 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.

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:`MultinomialHMM`, :class:`GaussianHMM`, 
and :class:`GMMHMM`. They implement HMM with emission probabilities
determined by multimomial distributions, Gaussian distributions
and mixtures 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 inferring the hidden states
-------------------------------------------------------

You can train an HMM by calling the `fit` method. The input is "the list" of 
the sequence of observed value. Note, since the EM algorithm is a gradient-based
optimization method, it will generally get stuck in local optima. You should 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 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 model2 will have a different order than those in the generating model.::

    >>> model2 = hmm.GaussianHMM(3, "full")
    >>> model2.fit([X]) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE
    GaussianHMM(algorithm='viterbi',...
    >>> Z2 = model2.predict(X)


.. topic:: Examples:

 * :ref:`example_applications_plot_hmm_stock_analysis.py`


Implementing HMMs with custom emission probabilities
----------------------------------------------------

If you want to implement other emission probability (e.g. Poisson), you have to
implement a new HMM class by inheriting the :class:`_BaseHMM` and overriding
the methods `__init__`, `_compute_log_likelihood`,
`_set` and `_get` for additional parameters,
`_initialize_sufficient_statistics`, `_accumulate_sufficient_statistics` and
`_do_mstep`.