Biomedical Engineering Reference
In-Depth Information
where the proof is presented in Sect.
5.7.3
.UsingEq.(
5.103
), we have
K
ʛ
−
1
tr
R
uu
ʨ
−
1
1
1
K
¯
AR
uu
¯
A
T
R
=
+
.
(5.104)
R
can be used in source imaging algorithms such as the adaptive beamformers.
This
5.4 Partitioned Factor Analysis (PFA)
5.4.1 Factor Analysis Model
The bioelectromagnetic data is often contaminated not only by sensor noise but also
by various types of interference of biological and nonbiological origins. A simple
modification of the VBFA algorithm enables the removal of such interferences. The
modified algorithm is called the partitioned factor analysis (PFA) [
1
]. The prerequisite
for the PFA algorithm is that a control measurement, which contains the interferences
but does not contain the signal of interest, be available. The factor analysis model
for PFA is expressed as
y
k
=
B
v
k
+
ʵ
for control data,
(5.105)
y
k
=
Au
k
+
B
v
k
+
ʵ
for target data.
(5.106)
In the equations above,
L
1 column vector
u
k
is the factor activity that represents
the signal of interest and
A
is an
M
×
×
L
mixing matrix. Also,
L
v
×
1 column vector
v
k
is the factor activity that represents the interference and
B
is an
M
×
L
interference
v
mixing matrix.
The PFA algorithm has a two-step procedure. The first step applies the VBFA
algorithm to the control data and estimates the mixing matrix
B
and the sensor-
noise precision
. The second step estimates the mixing matrix
A
and the signal
factor activity
u
k
. In the second step, the interference mixing matrix
B
and the noise
precision
ʛ
are fixed, and the VBFA algorithm is applied to the target data, which is
expressed as
ʛ
u
k
v
k
y
k
=
Au
k
+
B
v
k
+
ʵ
=[
A
,
B
]
+
ʵ
=
A
c
z
k
+
ʵ
,
(5.107)
where
u
k
v
k
A
c
=[
A
,
B
]
and
z
k
=
.
The equation above indicates that the second step can also be expressed by the factor
analysis model using the augmented factor vector
z
k
and the augmented mixing
matrix
A
c
.
