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

Marginal distributions are Poisson:
Y
i
~
i ij ik
Poisson(
θ + θ + θ ) ,
⎧i
= 1,..., n
⎪
⎨ jk , = 2,..., n.
⎪
⎩ jk , ≠ i
The joint probability function is given by
py ( , y, y; ) = PY [ = yY , = y, Y= y]
=
(2)
( y − x )
1 2
θ m
(3) m∈
( y
j
− xk)! xi!
.
x
A ∏ ∑ ∏
( j )
j∈R1 k∈R
i∈R
2
2
1 2 3 1 1 2 2 3 3
L
∑
∑
exp( − θ )
∏
j∈R
θ
j
∑
j k
( j )
k∈R
2
∏
i∈R
θ
xi
i
The calculation of above joint probability function is not easy. Here, we used recurrence
relations involving densities to compute the probability function. In 1967, Mahamunulu
presented some important notes regarding p variate Poisson distributions. According to
him the following recurrence relations are obtained for the trivariate two-way
covariance model:
y p( y , y , y ) = θ p( y − 1, y , y ) + θ p( y −1, y − 1, y ) + θ p( y −1, y , y − 1) (5.7)
1 1 2 3 1 1 2 3 12 1 2 3 13 1 2 3
y p( y , y , y ) = θ p( y , y − 1, y ) + θ p( y −1, y − 1, y ) + θ p( y , y −1, y − 1) (5.8)
2 1 2 3 2 1 2 3 12 1 2 3 23 1 2 3
y p( y , y , y ) = θ p( y , y , y − 1) + θ p( y −1, y , y − 1) + θ p( y , y −1, y − 1) , (5.9)
3 1 2 3 3 1 2 3 13 1 2 3 23 1 2 3
with py (
1, y2, y
3) = 0 if min{ y1, y2, y
3} < 0.
It also gives the following relations:
ypy
1
(
1, y2, 0) = θ1py (
1− 1, y2, 0) + θ12py (
1−1, y2− 1, 0) y1, y2
≥ 1
yp
2
(0, y2, y3) = θ2p(0, y2− 1, y3) + θ23p(0, y2−1, y3− 1) y2, y3
≥ 1
(5.10)
ypy
3
(
1,0, y3) = θ3py (
1,0, y3− 1) + θ13py (
1−1,0, y3− 1) y1, y3
≥ 1
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