Biomedical Engineering Reference
In-Depth Information
d
(
n
)
z
-1
z
-1
…
Wiener
filter
y
(
n
)
Subspace
projection
x
(
n
)
M
S
…
z
-1
z
-1
FIgURE 4.3:
The overall diagram of the subspace Wiener filter.
y
(
n
) denotes the estimated HP vector.
There are
L
− 1 delay operators (
z
−1
) for each subspace channel.
where
p
is defined as an
M
× 1 cross-correlation vector between
x
and
d
. The consecutive orthogonal
PLS projection vectors are computed using the deflation method [
14
].
There have been efforts to find a better projection that can combine the properties of PCA
and PLS. The continuum regression (CR), introduced by Stone and Brooks [
23
], attempted to
blend the criteria of LS, PCA, and PLS. Recently, we proposed a hybrid criterion function similar
to the CR, together with a stochastic learning algorithm to estimate the projection matrix [
24
]. The
learned projection can be either PCA, PLS, or combination of the two. A hybrid criterion function
combining PCA and PLS is given by
T
2
λ
T
1
−
λ
(
w p
)
(
w
w
)
R
s
(4.26)
J
(
w
,
λ
)
=
T
w w
where λ is a balancing factor between PCA and PLS. This criterion covers the continuous range
between PLS (λ = 1) and PCA (λ = 0).
2
Because the log function is monotonically increasing, the
criterion can be rewritten as,
2
(4.27)
(
)
(
)
T
T
T
w
log(
J
(
w
,
λ λ
))
=
lo
g
w
p
+ −
1
λ
) log(
w
w
)
−
log
w
R
s
We seek to maximize this criterion for 0 ≤ λ ≤ 1. There are two learning algorithms derived
in [
24
] to find
w
(one is based on gradient descent and the other is based on the fixed-point algo-
rithm), but we opted to use the fixed-point learning algorithm here because to its fast convergence
and independence of learning rate. The estimation of
w
at the
k
+1th iteration in the fixed-point
algorithm is given by
λ
p
+
−
(
1
λ
R
R
)
w
( )
k
(4.28)
(
1
)
(
1
)
( )
s
w
k
+ = −
T
w
k
+
T
T
T
w
( )
k
p
w
( )
k
w
( )
k
s
2
The CR covers LS, PLS, and PCA. However, because we are only interested in the case when subspace projection
is necessary, LS can be omitted in our criterion.
