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

nested in the other. Since, for instance, local independence model is nested in the
common covariance model by deleting the common interaction parameter; this therefore
seems like a reasonable test. However regularity conditions needed to use the LRT are
not satisfied, because the parameters that allow going from one model to the other take
a value at the boundary of the parameter space. Recall that the parameters of any
multivariate Poisson model are positive, so the value 0 is at the boundary. This makes
the use of the LRT statistic impossible. The same problem arises when testing for the
model fit between different component solutions and is well documented in the
literature (McLachlan and Peel, 2000).
Another solution for the goodness of fit of the model might be constructing some type
of information criterion, like the AIC and the BIC to test the difference between the
models. However, these information criterias compare point estimates and not the
difference between entire curves, so this does not seem to be applicable either.
Therefore, the one way of comparing the different solutions is by visually inspecting
loglikelihoods. Figure 6.12 indeed illustrates that the loglikelihood of the independence
and the common covariance models clearly dominate the loglikelihoods of the restricted
covariance model over the range of component solutions ( k =1 to 7). Figure 6.13
illustrates that the loglikelihood of the independence model clearly dominates the
loglikelihoods of the restricted and the common covariance model over the range of
component solutions ( k =1 to 7) for the Markov-dependent models. Viewpoints of the
model fit this partially justifies the use of the model with the independent covariance
structure since the comparison of maximized loglikelihood providing at least a rough
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