Biomedical Engineering Reference
In-Depth Information
3.
Completion:
N
(5.44)
P O
(
|
λ
)
=
=
1
α
( )
i
T
i
The computation of equation (
5.42
) in the induction step above (step 2) accounts for all
possible state transitions from time step t to time step
t
+ 1, and the observable at time step
t
+ 1.
Figure
5.6
below illustrates the induction step graphically.
The second problem is the optimization. To maximize the probability of the observation
sequence
O
, we must estimate the model parameters (
a
,
B
, π) for λ with the iterative Baum-Welch
method [
40
].
Specifically, for the Baum-Welch method, we provide a current estimate of the HMM λ =
[
a
,
B
, π] and an observation sequence
O
= [
O
1
, ... ,
O
T
] to produce a new estimate of the HMM
given by
{ }
=
,
B
,
A
λ
, where the elements of the transition matrix Ā
are,
T
−
1
∑
∑
( ,
)
ζ
i
j
t
1
}
t
=
,
,
{
1
,
...
,
N
a
=
i
j
∈
(5.45)
ij
T
−
1
.
γ
( )
i
t
t
=
1
Similarly, the elements for the output probability matrix
B
are given by,
∑
γ
( )
j
where
(
∀ =
O
v
)
t
t
k
t
b k
( )
=
,
j
∈
{
1
,...,
N k
}
,
∈
{ }
1
,...,
L
(5.46)
j
∑
=
T
γ
( )
j
t
t
1
FIgURE 5.6:
Forward update.
