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In-Depth Information
0.02
d
2
4
6
8
10
0
-0.02
-0.04
-0.06
Fig. 4.6
The function
Q
(
d
)
for the lognormal two-class problem.
4.1.3.1
Kernel-Based Estimates
Recall that
P
−
1
=
−∞
x
qf
X|−
1
(
x
)
dx
=
q
1
f
X|−
1
(
x
)
dx
,
x
−
(4.43)
−∞
P
1
=
p
x
−∞
f
X|
1
(
x
)
dx .
(4.44)
Using the KDE (3.2) to estimate
f
X|t
one gets
x
Φ
x
−
.
n
x
i
∈ω
t
1
x
i
f
X|t
(
x
)
dx
≈
(4.45)
h
−∞
An important difference with respect to the empirical version for continuous
errors is to be noticed. Here KDE is used to estimate the input PDFs in contrast
with the estimation of the error PDF performed in the previous chapter.
An appropriate optimization algorithm can now be used to obtain SEE so-
lutions. We illustrate this procedure with simulated data, namely two Gaus-
sian distributed classes satisfying the conditions of Theorem 4.2:
p
=
q
=1
/
2
and
σ
−
1
=
σ
1
. As discussed in the previous section, one may set
σ
t
=1and
vary the distance
d
between the classes. Given that
t
value
≈
1
.
405,weper-
formed experiments with generated data using
d
=3,
d
=1
.
5 and
d
=1,the
first two values corresponding to minimization problems and the third one to
a maximization problem. The experiments use different values of the training
set and test set sizes shown in Tables 4.1 and 4.2:
n
d
,n
t
instances, equally di-
vided between the two classes. Adequate values of
h
can be determined taking
formula (E.19) into account: for 100 instances per class,
h
IMSE
=0
.
42.
