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

The simple moments of B are polynomial with respect to the parameters
λ , t ∈{1,2,3,12,13,23}, and thus, the moments of the mixture can be obtained as
t
functions of the moments of the mixing distribution G using the standard expectation
argument given below.
r s
r s
E( Y Y ) = E(
Y Y | λ)
dG(
λ)
, (7.7)
i
j
where r,s = 0,1,…. The element-wise expectations of a matrix B (λ)
can be represented
∫
i
j
as:
2
⎡E( λ1 ) + E( λ1) E( λλ
1 2) ... E( λλ
1
) ⎤
t
⎢
2
⎥
E( λλ
1 2) E( λ2) + E( λ2) ... E( λλ
2 t
)
E( B ) = ⎢
⎥, (7.8)
⎢ ...
⎥
⎢
2
⎥
⎢⎣
E( λ1λt) ... E( λt ) + E( λt)
⎥⎦
then the unconditional variance of the vector Y (Karlis and Meligkotsidou, 2006), that
is the variance covariance matrix of the mixture is
Var ( Y)
Y)]
T
T
= E(
YY ) − E(
Y)[
E(
, where
E
T
T
( YY ) = AE(
B)
A . (7.9)
The moments of the multivariate Poisson distribution are simple polynomials with
respect to the mixing parameters. Comparing the estimated unconditional covariance
matrix to its observed covariance matrix can be used as a goodness of fit index.
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