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

q
In general, the number of parameters to be estimated is equal to ( k −1)
+ k × (2 −1)
,
for q varaites k -component mixture model. The number of parameters to be estimated
increases linearly in the number of components ( k ) and exponentially in the number of
the variables ( q ) being considered.
5.1.2 The multivariate Poisson model with common covariance structure
The fully structured multivariate Poisson model (section 5.1.1) has a large number of
parameters that need to be estimated. Therefore, an alternative approach has been
proposed in the literature to make a simpler version of the model by representing
variance/covariance by means of one common term (Johnson and Kotz, 1969, Li et al.,
1999 and Karlis, 2003).
In this approach, following the explanation in section 5.1, the trivariate Poisson variable
( Y1 , Y2
, Y3
) with one common covariance term, is defined as the following:
Y
Y
1
2
= X
1
= X
2
+ X
+ X
Y3
= X
3
+ X
123
with all X ’s independent univariate Poisson distribution with respective parameters θ
1,
θ , 2
θ
3,
and θ
123
.
123
123
In a matrix notation, this model can be presented as:
Y = AX
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