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

Using this equation we can calculate α ( j)
, 1≤ j≤ K , and then
T
K
P[ Y y ; λ] α ( j)
. (3.4)
= =∑
j=
1
T
This method is called the forward method and requires a calculation of the order K
2 T,
T
rather than 2TK , as required by the direct calculation previously mentioned.
As an alternative to the forward procedure, there exists a backward procedure (Baum et
al., 1967; Rabiner, 1989), which is able to solve P[ Y=
y ; λ]
. In a similar way, the
backward variable β (i)
can be defined as
t
*( t) *( t)
Y y
t
λ , (3.5)
β () i = P[ = | S = i; ]
t
where
*(t)
Y denotes { Y
t 1,
Yt
+ 2
,..., YT
}
+
(i.e. the probability of the partial observation
sequence from t+1 to T given the current state i and the model λ ).
Note that
β
*( T−1) *( T−1)
T−1 Y y
T−1
() i = P[ = | S = i; λ]
K
= PY [ = y ; S = i] =∑ Pb ( y ). (3.6)
T T T−1
ij j T
i=
1
As for of α ( j)
, one can solve for β (i)
inductively and can get the following recursive
relationship.
Now,
t
t
first initialize β ( i)
= 1,
1 ≤ i ≤ K.
(3.7)
T
Then for t = T −1,
T − 2,...,2, 1 and 1 ≤ i ≤ K,
*( t−1) *( t−1)
t−1 Y y
t−1
β () i = P[ = | S = i]
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