Biomedical Engineering Reference
In-Depth Information
E
u
log
p
u
K
)
Θ(
,
ʛ
)
=
(
y
1
,...,
y
K
,
u
1
,...,
A
E
u
log
p
u
K
)
+
E
u
log
p
u
K
)
=
(
y
1
,...,
y
K
|
u
1
,...,
(
u
1
,...,
K
K
=
E
u
log
p
(
y
k
|
u
k
)
+
E
u
log
p
(
u
k
)
,
(5.12)
k
=
1
k
=
1
where
E
u
[·]
indicates the expectation with respect to the posterior probability
. Taking a look at Eqs. (
5.4
) and (
5.7
), only the first term on the right-hand
side of Eq. (
5.12
) contains
A
and
p
(
u
|
y
)
ʛ
.Thus,wehave
K
log
p
u
k
)
Θ(
A
,
ʛ
)
=
E
u
(
y
k
|
+
C
k
=
1
K
K
2
1
2
E
u
T
=
log
|
ʛ
|−
1
(
y
k
−
Au
k
)
ʛ
(
y
k
−
Au
k
)
+
C,
(5.13)
k
=
where
C
expresses terms not containing
A
and
ʛ
.
The derivative of
Θ(
A
,
ʛ
)
with respect to
A
is given by
K
∂Θ(
A
,
ʛ
)
u
k
=
ʛ
E
u
1
(
y
k
−
Au
k
)
=
ʛ
(
R
yu
−
AR
uu
),
(5.14)
∂
A
k
=
where
K
K
u
k
u
k
u
k
ʓ
−
1
R
uu
=
E
u
=
1
¯
u
k
¯
+
K
,
(5.15)
k
=
k
=
1
K
K
y
k
u
k
u
k
,
R
yu
=
E
u
=
y
k
¯
(5.16)
k
=
1
k
=
1
R
yu
.
R
uy
=
(5.17)
Setting the right-hand side of Eq. (
5.14
) to zero gives
R
yu
=
AR
uu
. That is, the
M-step update equation for
A
is derived as
R
yu
R
−
1
A
=
uu
.
(5.18)
Next, the update equation for
ʛ
is derived. The partial derivative of
Θ(
A
,
ʛ
)
with
respect to
ʛ
is expressed as
T
K
∂Θ(
,
ʛ
)
∂
ʛ
A
K
2
ʛ
−
1
1
2
E
u
=
−
1
(
y
k
−
Au
k
)(
y
k
−
Au
k
)
.
(5.19)
k
=
