Digital Signal Processing Reference
In-Depth Information
2.5
1
2
0.8
1.5
0.6
1
0.4
0.5
0.2
0
0
-2 -1.5 -1 -0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5
0
0.5
1
1.5
2
(a)
u
(b)
u
FIGURE 7.10
Examples of one-dimensional RBFs with σ
=
1andμ
=
0: (a) Gaussian function and (b)
multiquadratic function.
1
0.8
0.6
0.4
0.2
0
10
0.8
0.6
0.4
0.2
10
5
10
5
10
5
5
0
0
0
0
-5
-5
-5
-5
-10 -10
-10 -10
(a)
(b)
FIGURE 7.11
Examples of a mapping built using an RBF network: (a) output of a single RBF neuron and
(b) output of an RBF network with 2 neurons.
approximation theorem similar to that discussed in the context of MLPs
[231].
Among the several possible choices, the classical option is for Gaussian
functions, so that, in the multidimensional case, we may have, for instance:
exp
2
−
x
(
n
)
−
μ
(
x
(
n
))
=
(7.49)
σ
2
where
μ denotes a mean vector that corresponds to a center
the variance σ
2
expresses a degree of dispersion around the center
The problem of finding the optimal parameters of RBF networks is some-
what different from that associated with the MLP network. In the case of
the MLP, all weights are “conceptually equivalent” in that they all belong
