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

the EM algorithms for use on very large data sets (McLachlan and Peel, 2000). Newton-
Raphson algorithm requires fewer iterations than the EM algorithm (McLachlan and
Basford, 1988). Quadratic convergence is regarded as the major strength of the Newton-
Raphson method. Furthermore, because of its computational straightforwardness, the
EM algorithm is the most extensively used (Titterington, 1990). Later in this chapter, a
detailed version of the EM for the multivariate Poisson finite mixture models and the
multivariate Poisson hidden Markov model is provided. At this moment, the EM can be
described as an iterative algorithm that sequentially improves on the sets of starting
values of the parameters, and facilitate simultaneous estimation of all model parameters
(Dempster et al., 1977; Hasselblad, 1969). More specifically, the observed data y i
is
augmented with the unobserved segment membership of subjects z ij
, which greatly
simplifies the computation of the likelihood instead of maximizing the likelihood over
the entire parameter space. More facts about the EM computation can be found in
Dempster et al. (1977) and McLachlan et al. (1988). The estimates of the posterior
probability
w
ij
, i.e. the posterior probability for subject i belongs to the component j ,
can be obtained for each observation vector y i
according to Bayes’ rule after the
estimation of the optimal value of Φ. In fact, after estimation the density distribution
f ( y i
| θ
j
) within each mixture component j and the component size p j
of each
component such that the posterior probability can be calculated as
w
ij
= k
p
∑
j=
1
j
p
f ( y
j
i
f ( y
| θ )
i
j
| θ )
j
. (5.15)
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