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

notation, the following recursive scheme is proved for the multivariate Poisson
distribution presented in (5.4):
yp( y) = θ p( y− e ) + θ p( y−1 ), i = 1,...,
n
(5.5)
i
i
i
0
k k
⎛ ⎞ θi
PY [
1
= y1,..., Yk = yk, Yk+
1
= 0,..., Yn
= 0] = p⎜y−∑e i ⎟∏
, for k = 1,..., n− 1, (5.6)
⎝ i= 1 ⎠ i=
1
yi
where the order of Y i
’s and 0’s can be interchanged to cover all possible cases, while
n
p( 0 ) exp( θ ). The recurrence equation (5.6) holds for arbitary permutations of Y i ’s.
= −∑
i=
0
i
It can be seen that since at every case at least one of the y i
’s equals 0, i.e. s = 0, the
sum appearing in the joint probability function has just one term and hence the joint
probability function takes the useful form P[ Y= y ] = exp( −θ
0) ∏ Po( yi; θi),
where
y
θ
Po(
y;
θ ) = exp( −θ ) denotes the probability function of the simple Poisson
y!
distribution with a parameter θ . Then equation (5.6) arises by using the recurrence
relation for the univariate Poisson distribution (Tsiamyrtzis and Karlis, 2004).
n
i=
1
Two examples of recurrence relations for the common covariance multivariate Poisson
distribution are given below. The θ in all recurrence relations is suppressed for
simplicity of the notation.
The bivariate Poisson distribution has joint probability function given by:
y1 y2
s
− ( θ0+ θ1+
θ2) θ1 θ2
⎛y1⎞⎛y2⎞
⎛ θ0
⎞
py (
1, y2) = PY [
1
= y1, Y2 = y2] = e ∑ ⎜ ⎟⎜ ⎟i! ⎜ ⎟ ,
y1! y2!
i=
0 ⎝i
⎠⎝i
⎠ ⎝θθ
1 2 ⎠
i
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