Biomedical Engineering Reference
In-Depth Information
and finally the
π
vector,
π γ
=
( ),
i
i
∈
{
1
,...,
N
.
}
(5.47)
1
where,
α
( )
i a
b
(
)
β
(
j
)
(5.48)
t
ij
j
t
+
1
ζ
( ,
i
j
)
=
t
+
1
t
(
)
P O
λ
and
N
å
1
g
( )
i
=
z
( ,
i
j
)
.
(5.49)
t
t
j
=
Note that β is the backward variable, which is similar to the forward variable α except that
now we propagate the values back from the end of the observation sequence, rather than forward
from the beginning of
O
[
40
]. First, we defined the backward variables (similar to the forward
variables) to,
(5.50)
β
( )
i
=
P O
(
,...,
O q
|
=
X
i
,
λ
)
,
t
t
+
1
T
t
which refers to probability of the partial observation sequence [
O
t+1
, … ,
O
T
] and being in state
X
i
at time
t
, given the model λ [
39
]. Just like the forward variables, the backward variables can be
computed inductively.
1.
Initialize:
(5.51)
β
T
( )
i
= ∈
1
,
i
{ ,...,
1
N
}
2.
Induction:
N
(5.52)
...
β
( )
i
=
∑
a
ij
b
j
O
(
)
β
(
j
),
t
∈ −
{
T
1
,
T
−
2
,..., },
1
i
∈
{ , ,
1
N
}
t
+
1
t
+
1
t
j
1
5.5.1 application of hMMs in BMIs
The goal of applying HMM to BMIs is different from the previously discussed generative models.
With HMMs the hypothesis is that the emerging states correspond to some underlying organiza-
tion of the local cortical activity that is specific to external events and to their behavioral signifi-
cance. Specifically, the HMM can be used to recognize a specific pattern (class) of motor neural
activity. If we assume that each class of neural data is associated with a given motor behavior, then a
