Biomedical Engineering Reference
In-Depth Information
where g is the number of group in the given samples and m
i
is not identical to m
j
.
In terms of S
B
and S
W
, the objective function J(
), called discriminant criterion, can
be written by
S
W
S
B
J
ðÞ¼
arg min
g
:
ð
4
:
16
Þ
This criterion introduced expects that within-class scatter value should be
minimized and the between-class scatter value should be maximized in the given
number. Under the minimizing Eq. (
4.16
), we can get the optimized number of
group (g) for RMLP by choosing the smallest J(
) with optimized group number
(g). This value can be used to test the optimized number of RMLP between HEKF
and DEKF. We can evaluate whether HEKF or DEKF could be more discriminated
by comparing the objective function values J(
) as the discriminant degree at the
selected (g).
4.3.4 Prediction Overshoot Analysis
Here, we evaluate the performance of overshoot for the prediction values. We
define overshoot for cases in which the predicted output exceeds a certain marginal
value with confidence levels corresponding to the tolerances. We would like to
derive such marginal value based on estimate process of the uncertainty point
estimators or predictors [
57
-
63
-
63
].
We noted in Eq. (
4.4
) in the previous Chapter that generally a neural network
model can be represented as a nonlinear regressive function as follows [
62
]:
y
i
ðÞ¼
U x
i
;
h
ð
Þþ
e
i
;
i
¼
1
;
2
;
...
;
n
;
ð
4
:
17
Þ
where x
i
(with dimension M 9 1) is input vector, and h (with dimension s 9 1) is
a set of neural network true weights. We assume that e
i
are independent and
identically-distributed with a normal distribution N(0, r
2
). Let define h as the least
square estimation of h. The linear Taylor series expansion for the model (
4.17
) can
be shown as follows [
63
]:
;
h
h
Þþ
U
0
y
i
ðÞ¼
U x
i
;
h
ð
i
¼
1
;
2
;
...
;
n
;
ð
4
:
18
Þ
where
:
U
0
¼
o
U
ð
x
i
;
h
Þ
o
U
ð
x
i
;
h
Þ
oh
2
oU
ð
x
i
;
h
Þ
oh
s
ð
4
:
19
Þ
oh
1
To construct marginal values for nonlinear regressive models in neural net-
works, the standard asymptotic theory should be applied. For the linear model in
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