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

Y
Y
Y
1
2
3
= X
= X
1
= X
2
3
+ X
+ X
+ X
12
12
13
+ X
+ X
+ X
13
23
23
+ X
+ X
+ X
123
123
123
Y = AX
A=[A 1 , A 2 , A 3 ]
⎡1
A =
⎢
⎢
0
⎢⎣
0
0
1
0
0
0
1
1
1
0
1
0
1
0
1
1
1⎤
1
⎥
⎥
1⎥⎦
⎡1 0 0⎤ ⎡1 1 0⎤ ⎡1⎤
where A1 =
⎢
0 1 0
⎥
,
⎢
2
1 0 1
⎥
,
⎢
3
1
⎥
⎢ ⎥
A =
⎢ ⎥
A =
⎢ ⎥
⎢⎣ 0 0 1⎥⎦ ⎢⎣0 1 1⎥⎦ ⎢⎣1⎥⎦
and X = X , X , X , X , X , X , ) .
(
1 2 3 12 13 23
X
123
The reduced models for n variables can be derived from selecting the A matrix. The
restricted covariance trivariate 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
⎡1 0 0⎤ ⎡1 1 0⎤
A1 =
⎢
0 1 0
⎥ ⎢
2
1 0 1
⎥
⎢ ⎥
A = .
⎢ ⎥
⎢⎣ 0 0 1⎥⎦ ⎢⎣0 1 1⎥⎦
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