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

The mean vector and the covariance matrix of the vector Y are given as:
T
E ( Y ) = AM and Var ( Y)
= AΣA
, (7.1)
where M is the mean vector of X and is given as:
M =
T
= E ( X)
( λ
1
,λ
2
,.., λ
k
) and
Σ is the variance and covariance matrix of X and is given as:
Σ = Var( X) = diag( λ1, λ2,..., λ
k)
. Since X ’s are independent, Σ is diagonal matrix.
More details and references for the multivariate Poisson model can be found in Karlis
and Xekalaki (2005). The identifiability and the consistency of finite mixtures of the
multivariate Poisson distribution with two-way covariance structure are proved in Karlis
and Meligkotsidou (2006).
In general notation, let f ( y;
λ)
∧ g(
λ)
be a general mixture of the density f (y;.)
with
λ
respect to its parameter λ , where
g(
λ),
λ ∈Θ
is the mixing distribution. The density of
the mixing distribution is given by f ( y)
= f ( y;
λ)
dG(
λ),
where G(λ)
is the
cumulative function of the mixing distribution.
∫
Θ
7.3 The properties of finite mixture models
The joint probability function of Y is p( y; λ ), and then the finite multivariate Poisson
mixture distribution can be given as:
k
f( y) = ∑ pjp( y; λ
j)
(7.2)
j=
1
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