multivariate poisson hidden markov models for analysis of spatial ...

For example, where the components correspond to externally existing groups is such
situation. Though, often the number of components has to be determined from the data,
along with the parameters in the component densities (McLachlan et al., 2000).
Regrettably, this crucial problem of finding the optimal number of components in a
mixture model has not yet been completely solved (Mackay, 2002). However, a more
suitable viewpoint to determine the number of components is based on the use of so
called information criteria. The most well-known examples include the AIC (Akaike
information criterion) (Akaike, 1974) and the BIC (Bayesian information criterion or
Schwarz information criterion) (Schwarz, 1978). The formulas are:
AIC = L k
− d k
d
k
BIC = Lk
− ln( n)
,
2
where L
k
- the value of maximized loglikelihood for a model with k components and
dk
- the number of free parameters in the model with k components and n is the number
of observations.
Information criteria are goodness of fit measures, which consider model parsimony. The
main idea is that the increase of the loglikelihood of the mixture model L k
on a
particular dataset of size n , penalized by the increased number of parameters d k
needed to create this increase of fit. A larger criterion indicates a better model in
comparison with another. In spite of this, it should be noted that several other criteria
exists. AIC and BIC criterions have been used to determine the number of states in a
hidden Markov model (Leroux and Puterman, 1992).
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