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6.5 Comparison of Radial Basis Function Network and Neuro-
fuzzy Network
There are considerable similarities, as well as dissimilarities, between the RBF-
type neural network and neuro-fuzzy network. In this section we present a few
comparisons between them.
A radial basis function network can be considered as a three-layer network
consisting of an input layer, a hidden layer and an output layer (see Chapter 3 for
details). The hidden layer performs the nonlinear transformation, so that the input
space is mapped into a new space. The output layer then combines the outputs of
the hidden layer linearly. The structure of an RBF network with an input vector
n
x \ and output y \ is shown in Chapter 3. The output from such a
network can be written as
N
y
x
¦
x
,
w R
i
i
i
1
where w are the weights and R i ( x ) is the nonlinear activation function of the
hidden-layer neurons.
The fuzzy logic system considered in Equations (6.9a) - (6.9c) can be rewritten
as
M
M
l
p
l
f
()
x
y
l
where
h
l
l
,
¦
h
z
¦
z
j
j
l
1
l
1
and
j
1, 2, 3,
"
,
m;
l
1, 2, 3,
"
,
M
.
noting that, when using the definition of the radial basis function the normalized
degree of fulfilment of the l th rule, i.e.
hh { , is similar to an RBF.
Therefore, the fuzzy logic system can also be represented as an RBF neural
network model. However, the following points have to be carefully noted:
l
l
i
x
Functions in the form of (6.9a) are just one kind of fuzzy logic system with
a particular choice of fuzzy inference engine with product inference rules ,
a fuzzifier, and a weighted-average defuzzifier. If another choice is made,
such as the mean-of-maxima (MOM) defuzzifier , then the fuzzy logic
system will be quite different from the RBF network. Therefore, an RBF
network in fact is a special case of the fuzzy logic system.
x
The membership functions of the fuzzy logic system can take various
geometric forms (such as Gaussian, triangular, trapezoidal, bell-shaped,
etc. ). They can also be non-homogeneous ( i.e. the membership functions
that divide the input or output universe of discourse may not all be of the
same functional form), whereas the RBF network takes a lesser number of
functional forms, like a Gaussian function, and are usually homogeneous.
This is due to the different justifications of the neuro-fuzzy network and
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