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

5.4 Multivariate Poisson hidden Markov models
Hidden Markov models (or Markov dependent finite mixture models) take a broad view
of mixture distributions by introducing serial correlation through the sequence of unseen
parameter values
λ
j
. In particular, this sequence is assumed to follow a Markov chain
with stationary transition probabilities. Formally, let { S i
} be a Markov chain with states
denoted 1,…,m and stationary transition probabilities. Then y i
are assumed to be
conditionally independent given S
i
, with conditional densities f y
i
; λ
S
) . To fit such a
(
i
model, the transition probabilities must be estimated along with the component
parameters
λ
j
. Details about the univariate hidden Markov model (or Markov
dependent finite mixture models) were described in Chapter 2 and 3.
5.4.1 Notations and description of multivariate setting
The following notations are used through out this section and do not refer to the
notations in other sections.
yij
= Measurement of the i th variable on the j th item.
⎡y y ... y ... y
11 12 1 j
1n
⎢
⎥
= ⎢ y21 y22 ... y2 j
... y2n
⎥
Y .
⎢y31 y32 ... y3 j
... y ⎥
⎣
3n⎦
⎤
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