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

Letting Ut( S1 = i1, S2 = i2,..., St = it)
be the first t terms of U (s)
and Vt( i
t)
be the minimal
accumulated distance when we are in state i at time t ,
U ( S = i , S = i ,..., S = i ) = − [ln( π b ( y )) +∑ ln( P b ( y ))]
t 1 1 2 2 t t S1 S1 1
Si−1S i Si
i
i = 2
t
V ( i ) = min U ( S = i , S = i ,..., S = i , S = i ) .
t t t 1 1 2 2 t−1 t−1
t t
S1, S2,... St−1,
St=
it
Viterbi algorithm can be carried out by following four steps:
1. Initialize the V1( i1)
for all 1 ≤ i≤
K :
V ( i ) =− ln( π b ( y )).
1 1
Si
Si
i1
2. Inductively calculate the Vt( i
t)
for all 1≤ it
≤ K , from time t = 2 to t = T :
V ( i ) = min [ V ( i ) − ln( P b ( y )].
t t 1 1
1
t − t −
≤i
S j S i S i i t
t−1
≤K
3. Then we get the minimal value of U (s) :
min U ( s ) = min[ V ( i )].
i1
, i2
,..., iT
T T
1≤iT
≤K
4. Finally we trace back the calculation to find the optimal state path
S
opt
= { S , S ,..., S
, opt 2, opt T ,
1 opt
}.
3.2.3 Problem 3 and its solution
This problem is concerned with how to determine a way to adjust the model parameters
so that the probability of the observation sequence given the model is maximized.
However, there is no known way to solve for the model analytically and maximize the
probability of the observation sequence. The iterative procedures such as the Baum-
Welch Method (equivalently the EM (expectation-maximization) method (Dempster et
37