Pattern Recognition Techniques - Biomedical Signal Analysis: Contemporary Methods and Applications - page 179

Δ w i = εe −k i ( x , w i /λ) ( x

− w i ) i =1 , ··· ,N, (6.19)

where i =1 , ··· ,N . The parameters have the time dependencies λ ( t )=

t
t max

and ε ( t )= ε i ( ε f /ε i ) t

λ i ( λ f /λ i )

t max
Increment the time parameter t by 1.
5. Continuation : Go to step 2 until the maximum iteration number t max
is reached.

6.4 Radial-Basis Neural Networks (RBNN)

Radial-basis neural networks implement a hybrid learning mechanism.
They are feedforward neural networks with only one hidden layer; their
neurons in the hidden layer are locally tuned; and their responses to an
input vector are the outputs of radial-basis functions. The radial-basis
functions process the distance between the input vector (activation)
and its center (location). The hybrid learning mechanism describes a
combination of an unsupervised adaptation of the radial-basis functions'
parameter and a supervised adaptation of the output weights using a
gradient-based descent method.
The design of a neural network based on radial-basis functions is
equivalent to model nonlinear relationships, and implement an interpo-
lation problem in a high-dimensional space. Thus, learning is equivalent
to determining an interpolating surface which provides a best match to
the training data. To be specific, let us consider a system with n inputs
and m outputs, and let

,y m }
the corresponding output vector describing the system's answer to that
specific input. During the training, the system learns the input and out-
put data distribution, and when this is completed, it is able to find the
correct output for any input. Learning can be described as finding the
“best” approximation function f ( x 1 ,

{ x 1 , ··· ,x n } be an input vector and { y 1 , ···

··· ,x n ) of the actual input-output

mapping function [70, 208].
In the following, we will describe the mathematical framework for
solving the approximation problem based on radial-basis neural net-
works. In this context, we will present the concept of interpolation net-
works and how any function can be approximated arbitrarily well, based
on radial-basis functions under some restrictive conditions.