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

clustering. For our data I (5) =0.7686 for the independent model, I (5) =0.7837 for the
common covariance model and I (4) =0.8568 for the restricted covariance model for
class of finite mixture models. In a similar manner, for the hidden Markov models, the
entropy criterions were I (5) =0.8425 for the independent model, I (5) =0.8119 for the
common covariance model and I (4) =0.8441 for the restricted covariance model. Both
classes of models indicate that the restricted covariance model had a very good
separation between components or states. Among these six models for the finite mixture
and the hidden Markov models, the entropy statistic is between 76%-85%. All models
can be considered, as “well separated” and the hidden Markov models had a very good
separation compared to the finite mixture models.
In the case of the multivariate finite mixture model, each multivariate observation can
be allocated to the clusters using the posterior probabilities. The multivariate observation
with the highest posterior probability in the k th cluster will be allocated to the
k th cluster. Figure 6.14 illustrates the contour plots of clusters for the independent, the
common and the restricted covariance multivariate finite mixture models.
Given the sequence of observations Y and the model with the transition probability
matrix, the most likely state sequence associated with the given observation sequence
can be found. This can be achieved by maximizing the probability of observing
observation sequence and the state sequence given their joint distribution. This can be
achieved using the so-called Viterbi Algorithm (Viterbi, 1967). After allocating each
observation to the corresponding states the optimal state path can be found. Figure 6.15
134