# 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.

## 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

- ensure that the character code for the parameter is missing from
`init_params`

and then - 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:

## Saving and loading HMM¶

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(algorithm='viterbi', ...
```

## 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

`base._BaseHMM._init` (X, lengths) |
Initializes model parameters prior to fitting. |

`base._BaseHMM._check` () |
Validates model parameters prior to fitting. |

`base._BaseHMM._generate_sample_from_state` (state) |
Generates a random sample from a given component. |

`base._BaseHMM._compute_log_likelihood` (X) |
Computes per-component log probability under the model. |

`base._BaseHMM._initialize_sufficient_statistics` () |
Initializes sufficient statistics required for M-step. |

`base._BaseHMM._accumulate_sufficient_statistics` (…) |
Updates sufficient statistics from a given sample. |

`base._BaseHMM._do_mstep` (stats) |
Performs the M-step of EM algorithm. |