Biomedical Engineering Reference
In-Depth Information
v
(
t
)
u
(
t
)
x
(
t
+
1
)
x
(
t
)
z
(
t
)
z
-1
C
+
+
e
(
t
)
A
System
K
t
+
Observer
~
~
t
x
(
t
+
1
x
)
~
t
z
)
z
-1
C
+
A
FIgURE 5.3:
Kalman filter block diagram.
also included the spike counts of
N
neurons in the state vector as
f
1
, … ,
f
N
. This specific formula-
tion would exploit the fact that the future hand position is not only a function of the current hand
position, velocity, and acceleration, but also the current cortical firing patterns. However, this ad-
vantage comes at the cost of large training set requirements, because this extended model would
contain many more parameters to be optimized. To train the topology given in Figure 5.3,
L
train-
ing samples of
x
(
t
) and
z
(
t
) are utilized, and the model parameters
a
and
U
are determined using
least squares. The optimization problem to be solved is (
5.15
).
-
∑
L
1
2
A
argmin
x
(
t
+
1
)
-
Ax
( )
t
(5.15)
=
A
t
=
1
The solution to this optimization problem is found to be (5.16)
T
(
T
1
(5.16)
A X X X X
=
)
-
1
0
1
1
where the matrices are defined as
=
[
. . .
]
=
[
. . .
]
. The estimate of the
x
x
,
x
x
X
X
0
1
L
−
1
1
2
L
covariance matrix
U
can then be obtained using (
5.13
).
T
=
-
U
(
X
-
AX X
)(
-
AX
)
/(
L
1
)
(5.17)
1
0
1
0
