Biomedical Engineering Reference
In-Depth Information
models for the empirical data:
p
em
(
y
)
=
p
I
(
y
)
+
ζ
(
y
)
,
(9.13)
where
ζ
(
y
) represent the error between
p
em
(
y
) and
p
I
(
y
). From Eq. 9.13,
ζ
(
y
)
can be rewritten as
ζ
(
y
)
=|
p
em
(
y
)
−
p
I
(
y
)
|
sign(
p
em
(
y
)
−
p
I
(
y
))
.
(9.14)
We assume that the absolute value of
ζ
(
y
) is another density which consists
of a mixture of normal distributions and we will use the following EM algorithm
to estimate the number of Gaussian components in
ζ
(
y
) and the mean, the
variance, and mixing proportion.
9.2.5.2 Sequential EM Algorithm
1. Assume the number of Gaussian components (
n
)in
ζ
(
y
)is2.
2.
The E-step
: Given the current value of the number of Gaussian compo-
nents in
ζ
(
y
), compute
δ
it
as
k
k
i
p
(
y
t
|
i
)
π
k
it
=
for
i
=
1to
n
and
t
=
1to
N
2
(9.15)
δ
l
=
1
π
l
)
,
.
l
p
(
y
t
|
3.
The M-step
: We compute the new mean, the new variance, and the new
proportion from the following equations:
t
=
1
δ
it
,
N
2
k
+
1
i
(9.16)
π
=
N
2
t
=
1
δ
k
it
y
t
k
+
1
i
(9.17)
µ
=
,
N
2
t
=
1
δ
k
it
N
2
t
=
1
k
i
)
2
k
it
(
y
t
−
µ
δ
k
+
1
i
)
2
(
σ
(9.18)
=
.
N
2
t
=
1
δ
k
it
4. Repeat steps 1 and 2 until the relative differences of the subsequent values
of Eqs. 9.16, 9.17, and 9.18 are sufficiently small.
5. Compute the conditional expectation and the error between
|
ζ
(
y
)
|
and the
estimated density (
p
ζ
(
y
)) for
|
ζ
(
y
)
|
from the following equations:
N
2
i
=
1
δ
it
ln
p
ζ
(
y
|
i
)
,
n
Q
(
n
)
=
(9.19)
t
=
1
n
(
n
)
=|
ζ
(
y
)
|−
1
π
i
p
ζ
i
(
y
)
.
(9.20)
i
=
