Biomedical Engineering Reference
In-Depth Information
We then rewrite Eq. (
4.37
)as
K
1
K
−
ʱ
j
[
ʓ
−
1
x
j
(
1
]
j
,
j
=
ʱ
j
1
¯
t
k
).
(4.40)
k
=
Thus, we can derive the following equation:
ʲ
ʓ
−
1
H
T
H
j
,
j
1
ʱ
j
=
.
(4.41)
K
k
=
1
¯
x
j
(
t
k
)
Equation (
4.41
) is called the MacKay update equation [
5
]. The MacKay update is
known to be faster than the EM update in Eq. (
4.38
) particularly when the estimation
problem is highly ill-posed, i.e.,
M
N
, although there is no theoretical proof that
guarantees the convergence of the MacKay update.
Let us summarize the algorithm to estimate the source distribution
x
k
.First,
<
ʱ
is set to an appropriate initial value and the parameters of the posterior distribution,
¯
x
k
and
ʓ
, are computed using Eqs. (
4.13
) and (
4.14
). Then, the hyperparameter
ʱ
is
updated using Eq. (
4.41
) with the values of
obtained in the preceding step.
The algorithm is similar to the EM algorithm and the only difference is the update
equation for
x
k
and
¯
ʓ
ʱ
.
4.5 Modified Algorithm Integrating Interference
Suppression
A simple modification of the algorithm described in the preceding sections leads to
an algorithm robust to the interference overlapped onto the sensor data
y
. The idea is
is that a control measurement, which contains the interference but not the signal of
interest, be available. Using the factor analysis model, the data is expressed such that
y
k
=
Bu
k
+
ʵ
for control data,
(4.42)
y
k
=
Hx
k
+
Bu
k
+
ʵ
for target data.
(4.43)
In the above equations,
L
×
1 column vector
u
k
is the factor activity and
B
is an
Similar to the PFA algorithm, this modified Champagne algorithm has a two-step
procedure. The first step applies the VBFA (or BFA) algorithm to the control data,
and estimates the interference mixing matrix
B
and the sensor-noise precision
M
×
ʲ
.
The target data is modeled as
x
k
u
k
y
k
=
Hx
k
+
Bu
k
+
ʵ
=[
H
,
B
]
+
ʵ
=
H
c
z
k
+
ʵ
,
(4.44)
