Biomedical Engineering Reference
In-Depth Information
TaBlE 4.5:
Online variable selection algorithm (taken from [30])
Given an
N
×
M
input matrix
X
(each row being
M
-dimensional sample vector), and
an
N
× 1 desired response matrix
y
, initialize
p
(0) = 0 and
R
(0) = 0
Transform
X
and
y
such that
1
N
1
N
1
N
å
=
for
j
= 1, … ,
M
.
Update the correlation:
p
(
n
) = (1 - ρ)
p
(
n
− 1)+ ρ(
d
(
n
)
x
(
n
))
Update the input covariance:
R
(
n
) = (1-δ)
R
(
n
− 1)+ δ(
x
(
n
)
T
x
(
n
))
å
=
,
å
=
, and
2
x
ij
0
x
ij
1
y
i
0
N
N
N
i
=
1
i
=
1
i
=
1
Let
c
(0) =
p
(
n
).
For
k
= 0, … ,
M
− 1
Let
C
max
=
max
j
c
{ }
, and
A
= [
j
: |
c
j
(k)| =
C
max
].
Compute a diagonal matrix
S
with elements of sign of
c
j
(
k
) for
j
∈
A
.
Φ =
SR
a
S
,
where
R
a
is submatrix of
R
(
n
) with
j
th rows and
j
th columns for
j
∈
A
.
α = (
1
a
T
Φ
−1
1
a
)
−1
θ
j
= α
R
acol
S
Φ
−1
1
a
,
where
R
acol
is a matrix consisting of
j
th columns of
R
(
n
) for
j
∈
A
.
j
C
−
−
c C
+
+
c
max
j
max
j
γ
=
min
+
,
Compute the step size,
.
C
j A
∈
α θ
α θ
j
j
Update
correlation:
c
j
(
k
) =
c
j
(
k
) - γ θ
j
.
Hence, using Φ obtained by
R
(
n
) and
S
, we can compute α and consecutively θ
j
for
j
∈
A
. This
modification removes the computation of the equiangular vector in step e in Table
4.4
, which is not
directly required for computing θ
j
or γ.
In this way,
R
(
n
) and
p
(
n
) are estimated with the current input-output samples and fed into
the modified LAR routine. Through this routine, a subset of input variables is selected with certain
threshold for the
L
1
norm of the coefficient vector. Therefore, it is possible to estimate which input
variables are more correlated with the current desired response at every time instance. The proce-
dure of this online variable selection is described in Table
4.5
.
4.2.3 Real-Time Input Selection for linear Time-variant MIMo Systems
In the linear MIMO system considered in this section, system outputs are assumed to be linearly
and causally correlated with the neuronal spatiotemporal pattern of input channels. The temporal
pattern of each input channel is represented by embedding input time series using a time delay line.
In our representation, only a discrete time series is considered. Hence, an input sample is delayed
