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

The population of interest thus consists of k subpopulations and the density (or
probability function) of the q -dimensional observation y from the j th ( j = 1,..., k)
subpopulation is f ( y | θ
j
) for some unknown vector of parameters θ
j
. The interest lies
in finding the values of the non-observable vector ϕ = φ , φ ,..., φ ) which contains the
(
1 2 n
component labels for each observation ( i = 1,...., n)
and φ j if the i th observation
belongs to the j th subpopulation.
i
=
Since the component labels are not observed, the conditional density of the vector y is
a mixture of density of the form
k
∑
f ( y ) = p f ( y | θ ) , (5.13)
i
j=
1
j
i
j
k
p j
j=
1
where 0 < p j
< 1, ∑
= 1and
p
j
are the mixing proportions. Note that the mixing
proportion is the probability that a randomly selected observation belongs to the j -th
component. This is the classical mixture model (McLachlan and Peel, 2000). The
purpose of model-based clustering is to estimate the parameters p ,..., , θ ,..., θ )
(
1
p k − 1 1 k
.
The maximum likelihood (ML) estimation approach, estimates the parameters
maximizing the loglikelihood:
n k
⎛
⎞
L( y;
θ , p)
= ∑ln⎜∑
⎟
p
j
f ( y i
| θ
j
)
. (5.14)
i= 1 ⎝ j=
1
⎠
But this is not easy since there is often not a closed-form solution for calculating these
parameters. Fortunately, due to the finite mixture representation, an expectation-
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