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

Y = X + X + X
1 1 12 13
Y = X + X + X
2 2 12 23
3
=
3
+ .
13
+
23
Y X X X
The joint probability function of an observation Y = Y , Y , ) is given (Karlis, 2004)
as
(
1 2
Y3
( y − x )
(2)
L
j∈R1 i∈R2
y θ =
1
= exp( )
1 2
=
2 3
=
3
= − θ
∑
m
( )! !
(3) m∈
yj − xk x ,
A ∑
x
∏ ∏ i
( j )
j∈R1 k∈R
i∈R
2
2
p( ; ) PY [ y , Y y , Y y ]
∑
∏
θ
j
∑
j k
k∈R
( j )
2
∏
θ
xi
i
where A = {1,2,3,12,13,23}. The unconditional probability mass function is given
under a mixture with k -components model by pp( y; θ ) = ppy (
1, y2, y3; θ ).
k
∑
k
∑
j j j j
j= 1 j=
1
As a result, the model assumes covariance between all the variables since it is imposed
by the mixing distribution. For a model with k components the number of parameters
equals to 7 k -1 (that is, 6 theta’s per component plus the mixing proportions), which,
compared to the fully saturated model that contains ( k −1)
+ k × (2 −1)
parameters,
increases linearly instead of exponentially with the number of components considered.
q
5.2 Computation of multivariate Poisson probabilities
The multivariate Poisson distribution is one of the well-known and important
multivariate discrete distributions. Nevertheless this distribution has not found a lot of
practical applications except the special case of the bivariate Poisson distribution
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