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

3.2.2 Problem 2 and its solution
Given a sequence of observations Y = { Y1, Y2,..., Y T
} and the model λ , we want to find
the most likely state sequence associated with the given observation sequence.
The solution to this problem depends on the way “the most likely state sequence” is
defined. One method is to find the most likely state S
t
at time t and to concatenate all
such S t
’s. However, sometimes this approach does not give a physically meaningful
state sequence. The most widely used criterion is to maximize P[ Y= y, S=
s ; λ]
. That is,
to maximize the probability of observing observation sequence Y = Y , Y ,..., Y } and
the state sequence S = { S1,..., S T
} given their joint distribution f ( ys , ).
{
1 2 T
Since the model λ = ( A,
B,
π ) and the observation sequence is Y = Y , Y ,..., Y }, the
{
1 2 T
probability of the state path and observation sequence given the model would be:
P[ Y= y, S= s; λ] = P[ Y= y| S= s; λ] P[ S=
s ; λ]
= π b ( y ) P b ( y )... P b ( y ). (3.10)
S1 S1 1 S1S2 S2 2 ST−1ST ST
T
To write this in a form of summations, we define U (s)
as
U ( s ) = − ln( P[
Y = y,
S = s;
λ])
T
=− [ln( π b ( y )) +∑ ln( P b ( y ))] . (3.11)
S1 S1 1
St−1St St
t
t 2
=
Since ln() is monotonic function if − ln( P[ Y= y, S=
s; λ])
is minimum, then this will give
us the state sequence for P[ Y= y, S=
s ; λ]
is maximum. Therefore,
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