Biomedical Engineering Reference
In-Depth Information
3.3.3.5
Deflation Algorithm
If more than one source are to be extracted, the above procedure may be run several
times. To extract each source only once, an idea is to remove from the observa-
tion the contribution of the sources already extracted. More specifically, denote
s
1
=
w
1
x
the first source extracted by the algorithm described in Sect.
3.3.3.4
.
Then, a new observation vector
x
1
can be built by removing the contribution of
s
1
as
de
=
x
−
h
1
s
1
,
x
1
(3.27)
where
h
1
is the regression vector minimizing the MSE (
3.7
) with respect to
h
,
whose solution is given by Eq. (
3.9
):
h
1
=
R
x
w
1
w
=
E
{
s
1
x
}
.
(3.28)
1
}
1
R
x
w
1
E
{
s
Now the rank of the covariance of
x
1
is not full anymore, because the subtraction
necessarily decreases it by one. Hence, the size of vector
x
1
can be diminished by
one without losing information. As explained in Sect.
3.2.2.3
, this dimensionality
reduction may be performed with the help of the PCA of
x
1
by retaining its
(
M −
1)
dominant principal components and neglecting the eigenvectors associated with null
eigenvalues. Including variance normalization as in Eq. (
3.18
), this operation can be
expressed as
P
1
=
D
−
2
1
de
=
P
1
x
1
T
x
1
with
U
1
,
(3.29)
where
D
1
and
U
1
contain the
(
M −
1)
nonzero eigenvalues and their corresponding
eigenvectors, respectively, of the covariance matrix of
x
1
. Vector
x
1
is now of
dimension
(
M −
1)
, is uncorrelated with the first extracted source
s
1
, and has
a full rank covariance matrix. The deflation procedure can thus continue until a
single source is left, when vector
x
M
is of dimension one. The complete deflation
algorithm is summarized in Algorithm
3
. In practical implementations of this
algorithm, mathematical expectations need to be replaced by, e.g., sample estimates
as defined in Eq. (
3.10
).
Improved deflation algorithm.
When a large number of sources are extracted,
it may happen that the same source is extracted several times despite the use
of regression. This is due to rounding errors that accumulate through successive
deflation stages, especially when processing short observation windows. In order
to cope with this problem, one solution consists of minimizing the MSE function
E
with respect to
H
k
∈R
L×k
,where
s
k
=[
s
1
, s
2
,...,s
k
]
T
are
the first
k
extracted sources. This minimization leads to replacing the regression
step (
3.27
)and(
3.28
), or line
4
in Algorithm
3
, by the alternative regression
equation:
{
x
−
H
k
s
k
2
}
de
=
x
−
T
k
}
T
x
k
k
}
−
1
s
k
.
E
{
xs
E
{
s
k
s
(3.30)
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