Biomedical Engineering Reference
In-Depth Information
Since the linear relationship in Eq. (
2.14
) holds, the probability distribution of the
sensor data
y
(
)
t
is expressed as
2
exp
2
1
1
p
(
y
)
=
−
2
y
−
Fx
,
(2.17)
2
M
/
(
2
ˀ˃
)
2
˃
where the explicit time notation
(
t
)
is omitted from the vector notations
x
(
t
)
and
for simplicity.
1
This
p
y
(
t
)
as a function of the unknown parameter
x
is called the likelihood
function, and the maximum likelihood estimate
(
y
)
x
is obtained such that
2
x
=
argmax
x
log
p
(
y
),
(2.18)
where log
p
is called the log-likelihood function. Using the probability distribu-
tioninEq.(
2.17
), the log-likelihood function log
p
(
y
)
(
y
)
is expressed as
1
2
log
p
(
y
)
=−
2
y
−
Fx
+
C,
(2.19)
2
˃
where
C
expresses terms that do not contain
x
. Therefore, the
x
that maximizes
log
p
(
y
)
is equal to the one that minimizes
F(
x
)
defined such that
2
F(
)
=
−
.
x
y
Fx
(2.20)
That is, the maximum likelihood solution
x
is obtained using
2
x
=
argmin
x
F(
x
)
:
where
F
=
y
−
Fx
.
(2.21)
F(
)
This
in Eq. (
2.20
) is referred to as the least-squares cost function, and the
method that estimates
x
through the minimization of the least-squares cost function
is the method of least-squares.
x
2.6 Derivation of the Minimum-Norm Solution
In the bioelectromagnetic inverse problem, the number of voxels
N
, in general, is
much greater than the number of sensors
M
. Thus, the estimation of the source vector
x
is an ill-posed problem. When applying the least-squares method to such an ill-
posed problem, the problem arises that an infinite number of
x
could make the cost
1
For the rest of this chapter, the explicit time notation is omitted from these vector notations, unless
otherwise noted.
2
The notation argmax indicates the value of
x
that maximizes log
p
(
y
)
which is an implicit function
of
x
.
