Biomedical Engineering Reference
In-Depth Information
Thus, we derive
∂
∂
y
k
)
=
¯
A
T
y
k
−
(
¯
A
T
s
ʛ
¯
A
s
+
ʨ
−
1
s
+
ʦ
)
log
p
(
z
k
|
s
ʛ
M
z
k
.
(5.147)
z
k
Consequently, the coefficient of
z
k
gives
ʓ
=
¯
A
T
s
ʛ
¯
A
s
+
ʨ
−
1
s
+
ʦ
.
M
(5.148)
The mean
¯
z
k
is obtained as
=
ʓ
−
1
L
T
¯
s
k
¯
¯
z
k
=
ʛ
y
k
.
(5.149)
¯
A
T
u
k
5.5.3.2 M-Step
The derivation for the M step is also very similar to the M step of the PFA-VBEM
algorithm in Sect.
5.4.3.2
. The estimation of
p
(
A
|
y
)
is based on
K
1
2
E
z
T
log
p
(
A
|
y
)
=−
1
(
y
k
−
Ls
k
−
Au
k
)
ʛ
(
y
k
−
Ls
k
−
Au
k
)
k
=
M
1
2
a
j
ʻ
j
ʱ
−
a
j
.
(5.150)
j
=
1
∂
∂
The derivative
A
log
p
(
A
|
y
)
is obtained as,
K
∂
∂
u
k
A
log
p
(
A
|
y
)
=
E
z
ʛ
1
(
y
k
−
Ls
k
−
Au
k
)
−
ʛ
A
ʱ
k
=
=
ʛ
R
yu
−
ʛ
LR
su
−
ʛ
(
R
uu
+
ʱ
).
A
(5.151)
The precision,
ʻ
j
ʨ
, is equal to the coefficient of
a
j
, and it is obtained as
ʻ
j
(
R
uu
+
ʱ
)
.
Hence we have
ʨ
=
(
R
uu
+
ʱ
).
(5.152)
¯
A
, i.e.,
Setting the derivative to zero gives
¯
A
R
uu
+
ʱ
)
−
1
LR
su
)
ʨ
−
1
=
(
R
yu
−
LR
su
)(
=
(
R
yu
−
.
(5.153)
