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

P
jk
n
∑
vˆ
jk
i=
2
= n m
∑∑
i= 2 l=
1
vˆ
( i)
jl
( i)
. (5.24)
These equations, similar to the equations 5.17 for the mixing proportions in a mixture
distribution, can be thought of as weighted empirical relative frequencies. The
maximizing values of
λ
j
are obtained exactly as for independent observations. The
algorithm is terminated when the changes in parameter estimates are small.
5.4.2.2 The forward-backward algorithm
The forward-backward algorithm is again an extension of univariate case (Chapter 2
and 3) to a multivariate case. The forward-backward algorithm is used to calculate the
conditional probabilities uˆ
j
( i)
and ˆ ( i)
. It is based on simple recursive formulae for
the forward variable
α ( i) = P[ y , y ,..., y , S = j]
and the backward variable
j 1 2 n i
v jk
( i) = P[ y ,..., y | S = j]
(5.25)
β
j i+
1 n i
which yield the quantities of interest by
α
j() i β
j() i α
j() i β
j()
i
uˆ j
() i = =
m
∑αl
( n)
α () i β () i
l
∑
j=
1
j
j
and
vˆ
jk
( i)
=
P
jk
f ( y ; λ
i
∑
l
k
) α
j
( i −1)
β
k
( i)
. (5.26)
α ( n)
l
The α (i)
and β (i)
are calculated recursively in i using following formulae:
j
j
95