Biomedical Engineering Reference
In-Depth Information
6. Repeat steps 2, 3, 4, and 5, and increase the number of Gaussian com-
ponents
n
by 1 if the conditional expectation
Q
(
n
) is still increasing and
(
n
) is still decreasing, otherwise stop and select the parameters which
correspond to maximum
Q
(
n
) and minimum
(
n
).
Since EM algorithm can be trapped in a local minimum, we run the above
algorithm several times and select the number of Gaussian components and
their parameters that give maximum
Q
(
n
) and minimum
(
n
).
After we determined the number of Gaussian components that formed
|
ΞΆ
(
y
)
|
,
we need to determine which components belong to class
1
, and belong to class
2
, and so on. In this model we classify these components based on the mini-
mization of risk function under 0-1 loss. In order to minimize the risk function,
we can use the following algorithm. Note that the following algorithm is writen
for two classes but it is easy to generalize to
n
classes.
9.2.5.3 Components Classification Algorithm
1. All Gaussian components that have mean less than the estimated mean for
p
I
1
(
y
) belong to the first class.
2. All Gaussian components that have mean greater than the estimated mean
for
p
I
2
(
y
) belong to the second class.
3. For the components which have mean greater than the estimated mean for
p
I
1
(
y
) and less than the estimated mean for
p
I
2
(
y
), do the following:
(a) Assume that the first component belongs to the first class and the
other components belong to the second class. Compute the risk
value from the following equation:
β
Th
R
(
Th
)
=
p
(
y
|
1
)
dy
+
p
(
y
|
2
)
dy
,
(9.21)
Th
ββ
where
Th
is the threshold that separates class
1
from class
2
. The
above integration can be done using a second-order spline.
(b) Assume that the first and second components belong to the first
class and other components belong to the second class, and from
Eq. 9.21 compute
R
(
Th
). Continue this process as
R
(
Th
) decreases,
and stop when
R
(
Th
) starts to increase.
