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

For example, consider the three components trivariate finite mixture model with
following restricted covariance structure.
Y
Y
Y
1
2
3
= X
= X
1
= X
2
3
+ X
+ X
+ X
12
12
13
+ X
+ X
+ X
13
23
23
⎡1
0 0 1 1 0 ⎤
A =
⎢
⎥
⎢
0 1 0 1 0 1
⎥
and
⎢⎣
0 0 1 0 1 1 ⎥⎦
p ,
1
p and p
2 3
are mixing proportions. The element wise expectation of matrix B (λ)
are E B ) , E B ) and E B ) respectively. Then
(
1
(
2
(
3
E ( B)
= p1E(
B1)
+ p2E(
B
2
) + p3E(
B3)
2
⎡E( λ11) + E( λ11 ) E( λλ
11 21) ... E( λλ
11 1)
⎤
t
⎢
2
⎥
E( λλ
11 21) E( λ21) + E( λ21) ... E( λλ
21 t1)
E( B
1)
= ⎢
⎥
⎢ ...
⎥
⎢
2
⎥
⎢⎣
E( λ11λt1) ... E( λt1) + E( λt1)
⎥⎦
2
⎡E( λ12) + E( λ12) E( λ12λ22) ... E( λ12λ
2)
⎤
t
⎢
2
⎥
E( λ12λ22) E( λ22) + E( λ22) ... E( λ22λt
2)
E( B
2)
= ⎢
⎥
⎢ ...
⎥
⎢
2
⎥
⎢⎣
E( λ12λt2 ) ... E( λt2 ) + E( λt2
) ⎥⎦
2
⎡E( λ13) + E( λ13) E( λ13λ23) ... E( λ13λ
3)
⎤
t
⎢
2
⎥
E( λ13λ23) E( λ23) + E( λ23) ... E( λ23λt
3)
E( B
3)
= ⎢
⎥
⎢ ...
⎥
⎢
2
⎥
⎢⎣
E( λ13λt3 ) ... E( λt3) + E( λt3)
⎥⎦
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