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

al., 1977) or gradient techniques can be used to estimate the model parameters. We will
describe the solution for this problem based on the Baum-Welch method.
Baum-Welch algorithm
To describe the Baum-Welch algorithm (forward-backward algorithm) one needs to
define two other variables in addition to the forward and backward variables defined
previously.
The first variable is defined as the probability of being in state i at time t , and in state
j at time t + 1, given the model and the observation sequence
ξ (, i j) = P[ S = i, S = j| Y=
y ; λ]
. (3.12)
t t t+
1
Using Bayes law and the independency assumption, the equation (3.12) can be written
as
PS [
t
= iS ,
t+
1
= j, Y=
y; λ]
ξt
(, i j)
=
P[ Y=
y; λ]
() t () t *() t *() t
t
Y y λ Y y
t+
1 t
PS [ = i, = ; ] P[ = , S = j| S = i; λ]
=
P[ Y=
y; λ]
() t () t *() t *() t
t
Y y λ
t+ 1 t
Y y
t+
1 t
PS [ = i, = ; ] PS [ = j| S = iP ] [ = | S = jS , = i; λ]
=
P[ Y=
y; λ]
() t () t *( t+ 1) *( t+
1)
t
= = λ
t+ 1= t
=
t+ 1= t+ 1 t+
1= λ =
PS [ i, Y y ; ] PS [ j| S iPY ] [ y | S j; ] P[ Y y | St
+ 1
; ]
=
= j λ
P[ Y=
y; λ]
(3.13)
and by the way that forward and backward variables are defined, we can use them to
write ξ ( i,
j)
in the form
t
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