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Subsections


Data structures

Poisson mixture model

A Poisson mixture models is defined by the two (1-D) arrays:
wght
Mixture weights (positive and sum to 1)
rate
Corresponding rates (parameters of the Poisson distributions)

Poisson HMM

A Poisson HMM is defined by
TRANS
The transition matrix of the hidden chain
rate
Corresponding rates (parameters of the Poisson distributions)

Negative binomial HMM

A negative binomial HMM is defined by
TRANS
The transition matrix of the hidden chain
alpha
Corresponding (positive) shape parameters
beta
Corresponding (positive) inverse scales
where the Negative Binomial distribution is such that

$\displaystyle \operatorname{P}(N = n) = \left(\begin{array}{c} n+\alpha-1\\
\...
...pha
\left(\frac{1}{1+\beta}\right)^n \qquad \mathrm{for} \quad n \in \mathbb{N}$

which has mean $ \alpha/\beta$ and variance $ \alpha(1+\beta)/\beta^2$. The negative binomial distribution can be viewed as a Poisson (continuous) mixture for which the rate follows a $ \operatorname{Gamma}(\alpha, \beta)$ distribution (this is the method used for simulating from the model in nbh_gen). If you don't know what the negative binomial distribution is, you should refer, for instance, to [8] or perhaps to [9].

Note on the initial distribution of HMMs in H2M/cnt

Contrary to what was the case for the H2M main functions, the H2M/cnt HMM functions are mostly intended to deal with ergodic models which are estimated from a single long observation sequence (whereas left-right HMMs such as those used in speech processing need to be trained using multiple observation sequences). With a single (long) training sequence, the initial distribution is a parameter that has little influence and that cannot be estimated consistently. Taking this into account, it is assumed that the initial distribution (usually called pi0 in the H2M functions is uniform (equal probabilities for all states of the model).


Next: Examples Up: The H2M/cnt extension: models Previous: Nomenclature

Olivier Cappé, Aug 24 2001