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

A=[A 1 , A 3 ]
⎡1 0 0 1⎤
A =
⎢
0 1 0 1
⎥
⎢ ⎥
⎢⎣
0 0 1 1⎥⎦
where
A
A
1
3
⎡1 0 0⎤
=
⎢
0 1 0
⎥
⎢ ⎥
⎢⎣0 0 1⎥⎦
⎡⎤ 1
=
⎢⎥
⎢⎥
1.
⎢⎥ ⎣⎦ 1
Although the covariance structure is limited compared to the general definition of the
multivariate Poisson, there are a convenient number of parameters to deal with.
Moreover, trivariate Poisson distribution PY [
1
= y1, Y2 = y2, Y3 = y3]
can now be
obtained as the marginal distribution from PY [
1
= y1 , Y2 = y2, Y3 = y2, X123 = x123]
as follows:
min( y1, y2, y3)
∑
PY [ = y, Y = y , Y = y] = PY [ = y, Y = y , Y = y, X = x ]. (5.3)
1 1 2 2 3 3 1 1 2 2 3 3 123 123
(3)
x = 0
Substituting the X’s for the Y’s in (5.3) results in:
(4)
1
=
1 2
=
2 3
=
3
= ∑ L
1
=
1 2
=
2 3
=
3 123
=
123
(3)
x = 0
PY [ y, Y y , Y y] P[ X x, X x, X x, X x ] with all X ’s
(4)
independent univariate Poisson distributions and = [ ]
thus:
L where L5 = min( y1, y2, y3)
,
L 5
1 1 2 2 3 3
(4)
L
∑∏
PY [ = y, Y = y , Y = y] = P[ X = ( y − x)] P[ X = x]
=
x
L
(3)
(4)
∑
= 0
e
∑
∏
j j l i i
(3)
x = 0 j∈R1 l∈R3
i∈R3
− ( θ + θ + θ + θ )
1 2 3 123
∏
j∈R
∏
j∈R
θ
( y − x )
j
j
∑
l
l∈R3
∏
i∈R
θ
1 3
.
( y − x )! x !
∑
∏
1 3
3
xi
i
j l i
l∈R
i∈R
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