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

5.2.2 The multivariate Poisson distribution with restricted covariance
The fully structured multivariate Poisson model has not found any real data applications
because of the complicated form of the probability function and the excessive structure
of the model. In this thesis, the restricted covariance structure, which explains only
pairwise covariances, is considered. The restricted covariance multivariate Poisson
model can be presented as follows:
Y
Y
Y
1
2
3
= X
= X
1
= X
2
3
+ X
+ X
+ X
12
12
13
+ X
+ X
+ X
13
23
23
A=[A 1 , A 2 ]
⎡1
A =
⎢
⎢
0
⎢⎣
0
0
1
0
0
0
1
1
1
0
1
0
1
0
1
1
⎤
⎥
⎥
⎥⎦
where
A
A
1
2
⎡1 0 0⎤
=
⎢
0 1 0
⎥
⎢ ⎥
⎢⎣0 0 1⎥⎦
⎡1 1 0⎤
=
⎢
1 0 1
⎥
⎢ ⎥
.
⎢⎣0 1 1⎥⎦
Details about the multivariate Poisson model with two-way covariance structure can
also be found in Karlis and Meligkotsidou (2005).
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