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

over the range of the data points. For each set of starting values, the algorithm was run
for 200 iterations without caring about any convergence criterion. Then, the set of initial
starting values with the largest loglikelihood was selected. The EM iteration were
continued with these selected initial values until the convergence criterion is satisfied,
i.e. until the relative change of the loglikelihood between two successive iterations was
smaller than
12
10 − . This procedure is repeated 7 times for each value of k .
The selection of number of clusters were based on the most well-known information
criterions (section 5.3.4), i.e. the Akaike Information Criterion (AIC) and the Bayesian
Information Criterion (BIC). The AIC is given as AIC= L k
− d k
and the BIC is given as
BIC= Lk
− ln( n) dk
/ 2 where L
k
is the value of the maximized loglikelihood for a model
with k components and d k
is the number of free parameters of the model. For the
restricted covariance, the independent and the common covariance models d k
is
d k
2
2
2
= 6 k + k − k , = 3 k + k − k and = 4 k + k − k respectively.
d k
d k
Figure 6.9 illustrates the evolution of the loglikelihood for the different components
(k=1,…,7) of the local independence multivariate Poisson hidden Markov model. This
figure illustrates that the AIC and the BIC selects five states as the optimal number of
states.
Similarly, Figure 6.10 and Figure 6.11 illustrate the evolution of the loglikelihood for
the different components of the common covariance and the restricted covariance
models respectively. Based on the AIC and the BIC criterion, the five states for the
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