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

where
p
j
’s are mixing proportions and the marginal distributions are finite mixtures.
Then the expectation of the finite multivariate Poisson mixture is given as:
for the reduced model.
k
∑ p j
j=
1
E( Y ) = AM , (7.3)
j
T
where M = λ ; t ∈ Ω,
Ω = {1,2,3,12,13,23}
j
tj
Different covariance structures can be formed for the different subpopulations by
changing the matrix A for each subpopulation.
Recalling that the covariance of X conditional on the vector λ (Karlis and
Meligkotsidou, 2006)
⎡λ1
0 ... 0 ⎤
⎢
⎥
⎢
0 λ2
... 0
Σ = Var(
X | λ)
=
⎥ , (7.4)
⎢ 0 .... .... 0 ⎥
⎢
⎥
⎣ 0 .... .... λt
⎦
where X denotes the vector of the latent variables used to construct the multivariate
Poisson distribution. The second moment of Y conditional on λ is given by
Let
B +
T
( λ)
= Σ MM and (λ)
T
T
E ( YY | λ)
= Var(
Y | λ)
+ E(
Y | λ)[
E(
Y | λ)]
T
= AΣA + AM( AM)
= AΣA + AMM A
T T T
T
= A[ Σ + MM ] A
B has the following form:
T
T
(7.5)
2
⎡λ1 + λ1 λλ
1 2
... λλ ⎤
1 t
⎢
2
⎥
λ1λ2 λ2 + λ2 ... λ2λt
B( λ ) = ⎢
⎥ . (7.6)
⎢ ... ... ... ... ⎥
⎢
2
⎥
⎢⎣
λ1λt ... ... λt + λt⎥⎦
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