State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 338

which can be used to approximate f ( t ) when the weights c are properly chosen.
Mukherjee and Nayer (1996) proposed a methodology for automatic generation
of RBF networks based on the integral wavelet transform . In fact, they concentrate
on automated construction of a generalized radial basis function network . To
solve the problem considered, there is a general question to be answered: Can a
multivariate function f ( x ) be represented by the sums and products of univariate
functions? The answer is to be found in approximation theory, which for this
purpose recommends minimizing the cost functional

f
f

> @ > @ 2

³

H FW x (,) f x () FW x (,) dx

with respect to W . In order to make the approximation problem well posed ,
regularization techniques have to be used by introducing smoothness constraints
into the approximation problem, so that the extended cost functional becomes

N

> @

¦

2
(,)

H FW x (,) [() f x FW x (, ]

2

+ O PF W x .

i i

i 1

Solving this problem (for details see Chapter 3), the approximation function for the
generalized radial basis network is defined by

n

¦ F Wx (,) cGxz (; ) ,

j j

j 1

where z , j = 1, 2, …, n , are the centres of the new basis functions, which can be
computed - along with the coefficients in the last equation - by minimizing the cost
functional

N

> @

¦

H FW x (,) [(,) FW x f x ( ]

2

.

i i

i 1

Based on the results of Zang (1997) in the use of wavelet network in non-
parametric estimation, Li and Chen (2002) proposed a robust wavelet network ,
based on the theory of robust regression .

10.3.3 Applications

As mentioned earlier, wavelets have been widely used in various application fields
of engineering. Some remarkable achievements have been reported in the
Proceedings of the IEEE , special issue on wavelets, in April 1996. A state-of-the
art report on wavelet applications in signal processing was compiled by Rioul and
Vetterly (1991). Also, Li et al . (2000) have presented a real-life application of the
wavelet transform in manufacturing for tool wear condition monitoring and tool