Biomedical Engineering Reference
In-Depth Information
4.8.3 Update Equation for
ʥ
j
The update equation for
ʥ
j
is obtained using
1
K
tr
Z
j
ʥ
j
K
ʥ
j
s
j
(
t
k
)
ʥ
−
1
=
argmin
ʥ
j
s
j
(
t
k
)
+
.
(4.83)
j
k
=
1
Taking
∂
∂
ʥ
j
s
j
(
t
k
)
ʥ
−
1
t
k
)
=−
ʥ
−
1
s
j
(
t
k
)
ʥ
−
1
s
j
(
s
j
(
t
k
)
(4.84)
j
j
j
into consideration, we have
1
K
tr
Z
j
ʥ
j
K
∂
∂
ʥ
j
s
j
(
t
k
)
ʥ
−
1
s
j
(
t
k
)
+
j
k
=
1
1
K
K
=−
ʥ
−
1
j
s
j
(
ʥ
−
1
j
s
j
(
t
k
)
t
k
)
+
Z
j
.
(4.85)
k
=
1
Setting the right-hand side to zero, we get the equation,
1
K
K
s
j
(
ʥ
j
Z
j
ʥ
j
=
s
j
(
t
k
)
t
k
)
.
(4.86)
k
=
1
ʥ
j
that satisfy Eq. (
4.86
).We should find
a positive semidefinite matrix that satisfies Eq. (
4.86
). Defining
However, there aremultiple solutions for
)
k
=
1
Ξ
=
(
1
/
K
s
j
(
s
j
(
t
k
)
t
k
),
and using
Z
−
1
/
2
j
Z
1
/
2
j
Z
1
/
2
j
Z
−
1
/
2
j
Ξ
=
(
Ξ
)
Z
−
1
/
2
j
Z
1
/
2
j
Z
1
/
2
j
Z
1
/
2
j
Z
1
/
2
j
2
Z
−
1
/
2
j
1
/
2
1
/
=
(
Ξ
)
(
Ξ
)
Z
−
1
/
2
j
Z
1
/
2
j
Z
1
/
2
j
2
Z
−
1
/
2
j
Z
j
Z
−
1
/
2
Z
1
/
2
j
Z
1
/
2
j
2
Z
−
1
/
2
j
1
/
1
/
=
(
Ξ
)
(
Ξ
)
,
(4.87)
j
the solution for
ʥ
j
that is a positive semidefinite matrix is derived such that
Z
1
/
2
j
1
K
Z
1
/
2
j
1
/
2
K
ʥ
j
Z
−
1
/
2
j
Z
−
1
/
2
j
s
j
(
=
s
j
(
t
k
)
t
k
)
.
(4.88)
k
=
1
Equation (
4.88
) is the update equation for
ʥ
j
.
ʥ
j
are updated using Eqs. (
4.79
), (
4.82
) and (
4.88
),
respectively. Since the auxiliary variable
x
k
is equal to the posterior mean of the
In summary,
Z
j
,
s
j
(
t
k
)
, and
