Biomedical Engineering Reference
In-Depth Information
Substituting Eq. (
2.58
)into(
2.57
), we get the cost function
2
2
F(
x
)
=
ʲ
y
−
Fx
+
ʱ
x
.
(2.59)
The cost function in Eq. (
2.59
) is the same as the cost function in Eq. (
2.37
), assuming
ʻ
=
ʲ/ʱ
. Thus, the solution obtained by minimizing this cost function is equal to the
solution of the
L
2
-norm regularized minimum-norm method introduced in Sect.
2.8
.
To obtain the optimumestimate of
x
, we should compute the posterior distribution.
In this case, the posterior is known to have a Gaussian distribution because
p
(
y
|
x
)
and
p
are both Gaussian, and the mean and the precision matrix of this posterior
distribution is derived as in Eqs. (B.24) and (B.25). Substituting
(
x
)
ʦ
=
ʱ
I
and
ʛ
=
ʲ
I
into these equations, we have
F
T
F
ʓ
=
ʱ
I
+
ʲ
,
(2.60)
F
T
F
I
−
1
+
ʱ
ʲ
F
T
y
x
¯
(
t
)
=
(
t
).
(2.61)
The Bayesian solution which minimizes the cost function in Eq. (
2.59
)isgivenin
Eq. (
2.61
). This solution is the same as Eq. (
2.39
). Comparison between Eqs. (
2.61
)
and (
2.39
) shows that the regularization constant is equal to
, which is the inverse
of the signal-to-noise ratio of the sensor data. This is in accordancewith the arguments
in Sect.
2.8
that when the sensor data contains larger amounts of noise, a larger
regularization constant must be used.
The optimumvalues of the hyperparameters
ʱ/ʲ
can be obtained using the EM
algorithm, as described in Sect. B.5.6. The update equations for the hyperparameters
are:
ʱ
and
ʲ
1
K
ʓ
−
1
K
tr
1
3
N
ʱ
−
1
x
T
=
1
¯
(
t
k
)
¯
x
(
t
k
)
+
,
(2.62)
k
=
1
K
ʓ
−
1
tr
F
T
F
K
1
M
ʲ
−
1
2
=
1
y
(
t
k
)
−
F
x
¯
(
t
k
)
+
.
(2.63)
k
=
Here, we assume that multiple
K
time-point data is available to determine
ʱ
and
ʲ
.
The Bayesian minimum-norm method is summarized as follows. First,
ʓ
and
¯
x
(
t
k
)
are computed using Eqs. (
2.60
) and (
2.61
) with initial values set to
ʱ
and
ʲ
.
Then, the values of
ʱ
and
ʲ
are updated using (
2.62
) and (
2.63
). Using the updated
ʱ
are updated using Eqs. (
2.60
) and (
2.61
). These
procedures are repeated and the resultant
and
ʲ
, the values of
ʓ
and
x
¯
(
t
k
)
.
The EM iteration may be stopped by monitoring the marginal likelihood, which
is obtained using Eq. (B.29) as
x
¯
(
t
k
)
is the optimum estimate of
x
(
t
k
)
K
1
2
K
log
1
2
y
T
t
k
)
ʣ
−
1
log
p
(
y
(
t
1
),...,
y
(
t
K
)
|
ʱ, ʲ)
=−
|
ʣ
y
|−
(
y
(
t
k
),
(2.64)
y
k
=
1
