Information Technology Reference
In-Depth Information
nh
3
K
(0)
I
which means
1
Thus, the Hessian matrix at
e
=
0
is of the form
nh
3
K
(0) with multiplicity
n
.The
Hessian is positive definite, and thus
R
ZED
has a maximum at
e
=
0
,if
K
(0)
<
0. We can also show that the maximum is global. First note that
1
that it has an unique eigenvalue
λ
=
f
E
(0)
e
=
0
≥
f
E
(0)
e
⇔
K
.
n
e
i
h
nK
(0)
≥
−
(5.14)
i
=1
If the kernel function is unimodal with mode at the origin, then
K
K
n
e
i
h
e
i
h
−
≤
n
max
i
−
≤
nK
(0)
.
(5.15)
i
=1
In short, if
K
(0) = 0,
K
(0)
<
0 and
K
is unimodal with mode at the origin,
then
R
ZED
attains its global maximum at
e
=
0
.
The Gaussian kernel satisfies the three properties of Theorem 5.1, and is thus
to be preferred for practical applications. Hence,
h
G
e
i
.
n
1
n
1
R
ZED
=
f
E
(0) =
(5.16)
h
i
=1
The kernel bandwidth (or smoothing parameter)
h
also plays an important
role in the success of this approach as illustrated in the following example.
Example 5.3.
Consider the two-class setting of Example 5.2. Let us analyze
the influence of
h
in estimating
R
ZED
. Figure 5.2 shows
f
E
(0) plotted as a
function of
w
0
for Gaussian generated input data (10 000 instances for each
class). Recall from Example 5.2 that
f
E
(0) is not defined, but here the KDE
produces a smoothed
f
E
(0) value. For small values of
h
(
h<h
IMSE
=0
.
27;
20
0.205
f
E
(0)
f
E
(0)
19
0.2
18
0.195
17
0.19
16
0.185
15
0.18
14
0.175
13
0.17
12
0.165
w
0
w
0
11
0.16
−5
0
5
−5
0
5
(a)
(b)
Fig. 5.2
f
E
(0)
plotted as a function of
w
0
for the two-class setting of Example 5.2
with two different values of
h
:a)
h
=0
.
01
;b)
h
=2
.
