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

loglikelihood providing at least a rough indication of the relative goodness of fit. The
same conclusion was gained after the primarily loglinear analysis and the correlation
matrix of the data.
In order to assess the quality of clustering, the entropy criterion was calculated based on
the posterior probabilities (McLachlan et al., 2000 and Brijs et al., 2004). A measure of
the strength of clustering is implied by the maximum likelihood estimates in terms of
the fitted posterior probabilities of component membership
w
ij
for the finite mixture
models and uj
( i ) for the hidden Markov models. For example, if the maximum of
or uj
( i ) is near to 1 for most of the observations, then it suggests that the clusters or
states were well separated (McLachlan et al., 2000). The overall measure of strength
can be assessed by the average of the maximum of the component-posterior
probabilities over the data. The average measure can be represented by the entropy
criterion given as
I(
k)
= 1−
n
k
∑∑
i= 1 j=
1
w
ij
ln( w
nln(1/
k)
for the finite mixture model with the convention that w ln( ) = 0 if w = 0 and
ij
)
ij
w ij
ij
w
ij
Im ( ) = 1−
n
m
∑∑
i= 1 j=
1
u ()ln( i u ()) i
j
nln(1/ m)
j
for the hidden Markov model with the convention that u ( i)ln( u ( i )) = 0 if u ( i ) = 0 . In
the case of perfect classification, for each i there is only one uj
( i ) = 1 and all the rest are
0 for the hidden Markov model: therefore, the values near to 1 indicate a good
j
j
j
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