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

where s = min{ y 1,
y 2}.
According to the general recurrence in (5.5) the following
recurrence are found:
ypy ( , y) = θ py ( − 1, y) + θ py ( −1, y −1)
1 1 2 1 1 2 0 1 2
ypy ( , y) = θ py ( , y− 1) + θ py ( −1, y−1),
2 1 2 2 1 2 0 1 2
with the convention that py (
1, y
2) = 0, if s < 0. Using these two recurrence relationships
interchangeably, one can get the entire probability table with
2
∏
i=
1
( y + 1) probabilities.
i
The trivariate Poisson distribution has joint probability function given by following:
y1 y2
y3
s
− ( θ0+ θ1+ θ2+
θ3) θ1 θ2
θ3 ⎛y1⎞⎛y2⎞⎛y
3 ⎞ ⎛ θ ⎞
0
py (
1, y2, y3) = PY [
1= yY
1, 2
= y2, Y3 = y3] = e ∑ ⎜ ⎟⎜ ⎟⎜ ⎟i! ⎜ ⎟ ,
y1! y2! y3!
i=
0 ⎝i
⎠⎝i
⎠⎝i
⎠ ⎝θθθ
1 2 3⎠
i
where s = min{ y1,
y2,
y3}.
Using the general recurrence in (5.5) the following
recurrence are found:
y p( y , y , y ) = θ p( y − 1, y , y ) + θ p( y −1, y −1, y −1)
1 1 2 3 1 1 2 3 0 1 2 3
y p( y , y , y ) = θ p( y , y − 1, y ) + θ p( y −1, y −1, y −1)
2 1 2 3 2 1 2 3 0 1 2 3
y p( y , y , y ) = θ p( y , y , y − 1) + θ p( y −1, y −1, y −1)
3 1 2 3 3 1 2 3 0 1 2 3
with the convention that py (
1, y2, y
3) = 0, if s < 0.
Tsiamyrtzis and Karlis (2004) demonstrated that how to use these existing recurrence
relationships efficiently to calculate probabilities using two algorithms called the Flat
and Full algorithms. This proposed algorithm can be extended to a more general
multivariate Poisson distribution that allows full structure with terms for all the pairwise
covariances, covariance among three variables and so on (Mahamunulu, 1967).
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