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

sequence. If we have several competing models, a solution to problem 1 allows us to
choose the model which best matches the observations.
A most straightforward way to determine P[ Y = y ; λ]
is to find out P[ Y = y, S=
s ; λ]
for a fixed state sequence S = { S1, S2,..., S T
} then multiply it by P[ S=
s ; λ]
and then
sum up over all possible states S .
We have a model λ and a sequence of observations Y = { Y1, Y2,..., Y T
} where T is the
number of observations and we want to find the probability of the observation sequence
P[ Y = y ; λ]
given the model. One could calculate P[ Y = y ; λ]
through enumerating
every possible state sequence of length T . Hence
∑
P[ Y= y; λ] = P[ Y= y| S= s; λ] P[ S=
s ; λ],
where S = { S1, S2,..., S T
}
∀S
= ∑ π b ( y ) P b ( Y )... P b ( y ).
(3.1)
S1, S2,...,
ST
S1 S1 1 S1S2 S2 2 ST−1ST ST
T
But this calculation for P( Y = y ; λ)
according to (3.1), involves several operations of
T
the order of 2TK , which is very large even if the length of the sequence, T , is
moderate. So another procedure must be applied to solve problem 1. Fortunately, this
procedure, the forward procedure, exists and calculates this quantity in a moderate time
(Baum et al., 1967; Rabiner, 1989).
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