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

seen that p values of the goodness of fit statistic of the models with some two-fold
interactions and without any interactions do not differ very much. We decided to
include all two-way interactions (section 6.2). Besides, two other interesting covariance
structures (common and independent structures which described in section 5.1.2. and
5.1.3) were considered.
9.2 Parameter estimation
Multivariate Poisson hidden Markov model is a special case of the hidden Markov
model. The estimation of the parameters of a hidden Markov model most efficiently has
done using the likelihood maximization. Baum and Eagon (1967) applied the EM
algorithm for locating a local maximum of the likelihood function for a probabilistic
function of a Markov chain. Baum et al. (1970) developed the EM algorithm, and
applied it to general hidden Markov model. The large-sample behaviour of a sequence
of maximum likelihood estimators for a probabilistic function of a Markov chain was
studied in Baum and Petrie (1966) and in Petrie (1969). Lindgren (1978) proved a
consistency property of maximum likelihood estimators obtained for the model, which
assumes that { Y i
} is an independent sequence from a finite mixture distribution.
Properties of the general ergodic hidden Markov models have been proven: the
consistency of the maximum likelihood estimators was proven by Leroux (1992a), and
the asymptotic normality of the maximum likelihood estimators was proven by Bickel
et al. (1998). Details of the maximum likelihood estimation of the hidden Markov
model are found in Leroux (1992 b).
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