Biomedical Engineering Reference
In-Depth Information
γ
ijk
=Cum(
z
i
,z
j
,z
k
,z
)=
μ
ijk
− R
ij
R
k
− R
ik
R
j
− R
i
R
jk
.
(3.22)
This cumulant indeed cancels out for all indices
(
i,j,k,
)
when
z
is made up of
Gaussian components.
Now under linear transforms
z
=
Wz
, moments and cumulants enjoy a
multilinearity property. For instance, the covariance matrix of
z
is related to that
of
z
by:
R
ij
=
mn
W
im
W
jn
R
mn
, with
W
im
=[
W
]
im
, or in compact form by
R
=
WRW
T
. Similarly, the cumulant of order 4 of
z
is related to that of
z
by
γ
ijk
=
mnpq
W
im
W
jn
W
kp
W
q
γ
mnpq
.
In particular, if we estimate one source from the observation
x
as
s
=
w
T
x
,
then the cumulant
γ
=Cum(
s, s, s, s
)
is related to the cumulants
C
mnpq
=
Cum(
x
m
,x
n
,x
p
,x
q
)
by
γ
=
mnpq
w
m
w
n
w
p
w
q
C
mnpq
,
(3.23)
where
w
m
=[
w
]
m
. With the help of these statistical tools, let us see first how we
can extract one source estimate
s
from the mixture
x
in Eq. (
3.17
).
3.3.3.2
Axiomatic Derivation
T
Our goal is to find a vector
w
such that
s
=
w
x
is close to one of the sources
s
i
.
T
=
w
T
In other words, the row vector
g
H
should contain a single nonzero entry;
such vectors are called
trivial filters
. To this end, we will maximize an objective
function
Ψ
(
w
)
, depending on
w
through
s
. It may hence be seen as a function
Ψ
[
s
]
of some feature of
s
, typically its probability distribution, which we distinguish with
brackets. According to Sect.
3.2.2.1
, PCA uses the output power (
3.4
) as an objective
function, which is maximized subject to the constraint
=1
. But we have seen
in Sect.
3.3.2
that, depending on the mixing matrix structure, the maximization of
this function is not always successful in extracting one of the sources. To perform a
successful source extraction in the noiseless case, the optimization criterion
Ψ
(
w
)=
Ψ
[
w
w
T
T
x
]=
Ψ
[
g
s
]
should satisfy the following properties, which may be viewed as
axioms:
1. For any trivial filter
t
, there should exist one source
s
i
such that
T
Ψ
[
t
s
]=
Ψ
[
s
i
]
.
This means that the objective function
Ψ
should be insensitive to trivial filters.
We can refer to this property as
invariance
.
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