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

5.2.1 The multivariate Poisson distribution with common covariance
Suppose that
X i
are independent Poisson random variables with mean θ i
, for
i = 0,1,...,
n and let Yi = X
0
+ X
i
, i = 1,...,
n . Then the random vector Y = ( Y1,
Y2,...,
Yn
)
follows a n-variate Poisson distribution, where n denotes the dimension of the
distribution. The joint probability function (Karlis, 2003) is given by
p( y; θ ) = PY [
1
= y1, Y2 = y2,..., Yn
= yn]
⎡ ⎛
⎢
θ ⎛ y ⎞ ⎜
= exp − ⎜ ⎟ !
n n yi
s n
⎛ ⎞
i
j
0
θ
⎢
i
i ⎥
⎜ ∑ ⎟∏
∑
⎜ ⎟
n
i= 1 i= 1 yi
! ⎢∏
⎝ ⎠ i=
0 j=
1 ⎝i
⎠ ⎜ ⎥
θ
⎟
⎢ ⎜∏
k ⎟ ⎥
k = 1
⎢⎣
⎝
θ
i
⎞ ⎤
⎟ ⎥
, (5.4)
where s = min{ y1,
y2,...,
yn}
. Marginally each Y i
follows a Poisson distribution with
parameter θ
0
+θ
i
. The parameter θ 0
(common covariance) is the covariance between
all the pairs of random variables ( Y, Y ) where i, j∈ {1,..., n}
and i ≠ j. If θ 0 then
i
j
the variables are independent and the multivariate Poisson distribution reduces to the
product of independent Poisson distributions. The recurrence relations can be applied to
compute the above probabilities. A general scheme for constructing recurrence relations
for multivariate Poisson distributions was provided by Kano and Kawamura (1991).
Details are given below.
⎠
⎥⎦
0
=
Some notations are introduced first to make it easy to explain the distributions. Let 0
and 1 denote the vector with all elements equal to 0 and 1 respectively and e
i
the unit
vector with all elements 0 except from the i th element which is equal to 1. Using this
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