Biomedical Engineering Reference
In-Depth Information
Also,
R
v
u
and
R
uu
are obtained in the following manner. We first define
ʓ
uu
ʓ
u
v
ʓ
v
u
ʓ
−
1
=
,
(5.126)
ʓ
vv
and have
K
R
uu
R
u
v
R
v
u
R
vv
u
k
v
k
u
k
,
v
T
=
E
z
[
k
]
k
=
1
k
=
1
¯
K
ʓ
uu
ʓ
u
v
ʓ
v
u
u
k
k
=
1
¯
T
k
u
k
¯
u
k
v
=
+
.
(5.127)
k
=
1
v
k
¯
u
k
k
=
1
v
k
v
ʓ
vv
T
k
Thus, we obtain,
K
K
ʓ
uu
,
u
k
R
uu
=
1
¯
u
k
¯
+
(5.128)
k
=
K
u
k
K
ʓ
v
u
.
R
v
u
=
1
v
k
¯
+
(5.129)
k
=
The hyperparameter
ʱ
is obtained by maximizing the free energy, which is
expressed as
F
=
E
(
A
,
u
)
[
log
p
(
y
|
z
,
A
c
)
+
log
p
(
z
)
+
log
p
(
A
c
)
−
log
p
(
z
|
y
)
−
log
p
(
A
c
|
y
)
]
.
(5.130)
However, since
ʱ
is contained only in log
p
(
A
c
)
, (which is equal to log
p
(
A
)
), the
update equation for
ʱ
is exactly the same as that in Eq. (
5.79
).
5.4.4 Summary of the PFA Algorithm
The PFA algorithm is summarized as follows. The first step estimates the interference
mixing matrix
B
and the diagonal noise precision matrix
by applying the VBFA
algorithm to the control data. The second step applies the PFA-VBEM algorithm to
the target data, and estimates the signal factor vector
u
k
and the signal mixing matrix
A
. In the second step,
B
and
ʛ
are fixed at the values obtained in the first step. There
is a different version of the PFA algorithm in which
B
and
ʛ
ʛ
are also updated in the
second step. The details of the algorithm are given in [
1
].
The free energy is computed in exactly the same manner as in Eq. (
5.100
) except
that
u
k
is replaced by
¯
¯
z
k
. Thus, the free energy is expressed as
