Biomedical Engineering Reference
In-Depth Information
On the other hand, according to Sect.
4.10.2
,
is expressed as
K
1
2
y
k
ʣ
−
1
=
y
k
,
(4.29)
y
k
=
1
where
ʣ
y
is given in Eq. (
4.25
). Thus, substituting Eqs. (
4.28
) and (
4.29
), into (
4.24
),
we get
K
1
2
K
log
1
2
y
k
ʣ
−
1
log
p
(
y
|
ʱ
)
=−
|
ʣ
y
|−
y
k
.
(4.30)
y
k
=
1
The above equation indicates that
p
(
y
|
ʱ
)
is Gaussian with the mean equal to zero
and covariance matrix equal to
ʣ
y
.This
ʣ
y
is called the model data covariance.
Therefore, the estimate of
ʱ
,
ʱ
, is obtained by maximizing log
p
(
y
|
ʱ
)
expressed
above. Alternatively, defining the cost function such that
K
1
K
y
k
ʣ
−
1
F(
ʱ
)
=
log
|
ʣ
y
|+
y
k
,
(4.31)
y
k
=
1
the estimate
ʱ
is obtained by minimizing this cost function.
4.4 Update Equations for
ʱ
ʱ
In this section, we derive the update equation for
. As will be shown, the updated
equation contains the parameters of the posterior distribution. Since the value of
ʱ
is needed to compute the posterior distribution, the algorithm for computing
is
a recursive algorithm, as is the case of the EM algorithm presented in Sect. B.5 in
the Appendix. That is, first setting an initial value for
ʱ
, the posterior distribution
is computed. Then, using the parameters of the posterior distribution,
ʱ
ʱ
is updated.
These procedures are repeated until a certain stopping condition is met.
Let us derive the update equation for
ʱ
byminimizing the cost function
F(
ʱ
)
, i.e.,
ʱ
=
argmin
ʱ
F(
ʱ
).
The derivative of
F(
ʱ
)
with respect to
ʱ
is computed,
K
∂F(
ʱ
)
∂ʱ
j
∂
∂ʱ
j
∂
∂ʱ
j
1
K
y
k
ʣ
−
1
=
log
|
ʣ
y
|+
y
k
.
(4.32)
y
k
=
1
The first term in the right-hand side is expressed using Eq. (
4.28
)as
∂ʱ
j
−
|
ʓ
|
∂
∂ʱ
j
∂
log
|
ʣ
y
|=
M
log
ʲ
−
log
|
ʦ
|+
log
=−
∂
∂ʱ
j
∂
∂ʱ
j
log
|
ʦ
|+
log
|
ʓ
|
.
(4.33)
