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hidden unit
j
. In the last equation, the factors
a
j
and
b
j
represent the
dilation
and the
translation
coefficients of the wavelet in the hidden layer respectively. Similarly,
j
w
represents the connecting weight between the hidden unit
j
and the input unit
k
. Relying on the above representation of neural networks, Cybenko (1989) and
Hornik
et al
. (1989) proved - using the Stone-Weierstrass theorem - that any
arbitrary function can be approximated with a given accuracy, thus designating the
single hidden-layer neural network as a
universal approximator
.
The proposed wavelet neural network is trained using the backpropagation
algorithm with the cost function
1
2
PN
¦¦
E
d
p
y
p
,
i
i
pi
11
where
d
is the desired network output of
p
th input pattern. Furthermore,
P
represents the sum of input sample and
m
,
n
, and
N
the sum of input, hidden, and
output nodes respectively.
Pati and Krishnaprasad (1993) developed an alternative structure of
feedforward network, based on the
discrete affine wavelet transform
. This is
possible because the sigmoid activation function can be viewed as being composed
of
affine wavelet decompositions
of mappings.
Zhang
et al.
(1995) described a wavelet neural network structure similar to that
of a radial basis function network in which the radial basis functions are replaced
by orthonormal scaling functions that are not necessarily radially symmetric. The
wavelets used for network implementation are functions whose translations and
dilations build an orthonormal basis of
L
2
(
R
), which encompasses all square
integrable functions of
R
, with the
mother wavelet
of the form
m
/2
m
\
mn
t
,
()
2
\
(2
t
n
).
The objective of the proposed network is that, given a training data set
^
`
Tt
,()
f t
,
N
i
i
where
i
=1, 2, …,
N
, the optimal estimate of
f
(
t
) could be found using
¦
ft
()
f
,
MM
()
t
.
Mk
,
Mk
,
k
For a given set of
M
and
k
, the wavelet network implements the mapping
K
¦
g t
()
c
M
()
t
kMk
kK
,
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