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

p values of the goodness of fit of the models with some two-fold interactions and
without any interaction do not differ very much. Therefore, the latent variables
X = X , X , X , X , X , ) are decided to keep in the model (i.e. use all two-fold
(
1 2 3 12 13
X
23
interaction terms). The vector of parameters is now θ = ( θ1, θ2, θ3, θ12, θ13, θ23)
and thus
the following restricted covariance model can be formulated:
Y
Y
Y
1
2
3
=
=
=
X
X
X
1
2
3
+
+
+
X
X
X
12
12
13
+
+
+
X
X
X
13
23
.
23
6.4 Data analysis
In this section, the computational results of the multivariate Poisson finite mixture
model and the multivariate Poisson hidden Markov model with the restricted covariance
structure is discussed and compared with the results of the local independence model
and the common covariance structure. The computational results of the fully saturated
multivariate Poisson finite mixture model and the fully saturated multivariate Poisson
hidden Markov model will not be discussed since there is no available method to
estimate the parameters of the fully saturated multivariate Poisson model in a reliable
way. As mentioned in section 5.1, the computation of the fully saturated model involves
a great number of summations and parameters to be estimated and this remains a
difficulty for calculation of the probability function. As a result, a comparison with the
fully saturated covariance model cannot be made.
114