Biomedical Engineering Reference
In-Depth Information
Layer 1
(Gaussian centers)
Layer 2
(Feed forward connections)
Network
inputs
x
p
Network
outputs
y
p
x
1
y
p
x
2
y
p
n
x
(
w
j
)
Network parameters:
(
1
w
j
,
j
)
FIGURE 5.3
The computational structure of a radial basis function (RBF) network. The input is the data vector
x
p
and the out-
put is the response given by vector
y
.
different definitions for the basis function have been proposed in the recent literature but
most nonsymmetrical approaches are application dependent (14).
Each node in the second layer of the RBF network computes a weighted linear summa-
tion of the response produced by the basis functions in the previous layer, and its output
is given by
N
2
y
i
u
1
j
w
2
i
(5.4)
i
0
where
y
i
is the output response of the
i
th
node in layer 2,
w
2
j
is the weight vector for this
node,
u
1
i
is the output from the
i
th
neuron in the first layer, and
N
2
is the number of output
nodes. The second layer of the RBF network also uses a bias input of
u
0
1 to adjust the
approximation curve or classification boundary. Thus, the overall network performs a
nonlinear transformation from
N
2
by forming a linear combination of the nonlin-
ear basis functions. Theoretically, the RBF network can form an arbitrarily close approxi-
mation to any continuous nonlinear mapping operation. Since the individual basis
functions of the RBF network cover small localized regions, the accuracy of the mapping
is related to the number of functions used in the network.
A variety of different training algorithms are used to determine the location of the
Gaussian centers (
w
1
j
), normalization parameters (
N
1
to
ℜ
ℜ
j
), and weights associated with the out-
put layer (
w
i
). Most algorithms begin by separating the problem into two stages:
unsu-
pervised
clustering to determine Gaussian centers and normalization parameters, and
supervised
least-mean square (LMS) adaptation of weights in the output layer. There are a
number of different clustering algorithms that can be used but the most popular choice is
the
K
-means clustering algorithm (14). The normalization parameters are obtained once
the
K
-means algorithm has completed the process of determining the kernel centers,
w
1
j
.
These values represent the spread of data associated with each node in layer 1
(i.e., Gaussian center). The most common technique for problems with large datasets is to
