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

notation of Marshall and Olkin (1985), and Johnson et al. (1997), Brijs (2002) and Brijs
et al. (2004), and based on the discussion in section 5.1, a trivariate Poisson random
variable ( Y
1,
Y2
, Y3
) with parameters ( θ1, θ2, θ3, θ12, θ13, θ23, θ
123)
can then be constructed
from a number of independent univariate Poisson distributions as follows:
Y
Y
Y
1
2
3
= X
1
= X
= X
2
3
+ X
+ X
12
+ X
12
13
+ X
+ X
13
+ X
23
23
+ X
+ X
+ X
123
123
123
with all X ’s are independent univariate Poisson distributions with their respective
means θ
1,
θ
2,
θ , θ
12,
θ , ,
3
13
θ
23
θ
123
. The calculation of the probability distribution of
PY [ = y, Y = y , Y = y]
is not easy. The solution is based on the observation that
1 1 2 2 3 3
PY [ = y, Y = y, Y = y]
is the marginal distribution from
1 1 2 2 3 3
PY [ = y, Y = y , Y = y, X = x , X = x , X = x , X = x ] and can be obtained
1 1 2 2 3 3 12 12 13 13 23 23 123 123
by summing out over all X ’s, i.e.,
PY [ = y, Y = y , Y = y]
=
1 1 2 2 3 3
L1 L2 L3
L4
∑∑∑∑
x12 = 0x13= 0x23= 0 x123=
0
PY [ = y, Y = y , Y = y, X = x , X = x , X = x , X = x ]
1 1 2 2 3 3 12 12 13 13 23 23 123 123
(5.1)
with L
1
= min( y1,
y2
),
L
= min( y1
− x12
,
3),
2
y
L = min( y −x , y −x
),
3 2 12 3 13
L = min( y −x −x , y −x −x , y −x −x
).
4 1 12 13 2 12 23 3 13 23
The above expression (5.1) demonstrates that the x’s are summed out over all possible
values of the respective X’s. It is known that the X’s should take only positive integer
values or zero, since the X’s are Poisson distributed variables. However, the upper
bounds (L’s) of the different X’s are unknown and will depend on the values of y
1,
y
2
,
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