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

• If
*
R = R 1
then the model reduces to an independence model (referred to as the
local independence model).
• If
= ∪ then the model reduces to a model with one common covariance
*
R R1 R3
term (referred to as the common covariance model).
• If the model assumes that
= ∪ then it allows only two-way covariances
*
R R1 R2
(referred to as the restricted covariance model).
Note that omitting the set of parameters θ ( j ∈ ), is equivalent to setting θ = 0 .
j
The submodels can be formed by assuming that the corresponding θ ’s equal zero. Now,
denote the cardinality of R as J which, for a trivariate model, equals J =7. Then, using
the above notation, and considering the most general model with all the covariance
terms (though it imposes unnecessarily large structure), the joint probability density of
the corresponding multivariate Poisson distribution is given as
R m
j
p( y | θ ) = PY [ = y, Y = y , Y = y | θ , j∈R]
1 1 2 2 3 3
J
= ∑.... ∑∏ Po( x | θ ) ,
where the summation is extended over all the combinations of
j∈R
j
j
j
x
j
such that
∑
y ≥ ,
i
x k
k
k ∈ R and k contains the subscript i . The fully-structured covariance model which is
illustrated in section 5.1.1 needs four summations for the trivariate case, which
obviously implies a large computational burden. The major problem of the use of the
probability distribution in its general form is the calculation difficulty of the probability
mass function. Kano and Kawamura (1991) described recursive schemes (section 5.2) to
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