State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 333

x determination of the optimal separation hyperplane that maximizes the
margin, i.e . the distance to the points.
We also recall that the adatron algorithm is capable of maximizing the margin.
This can be used to implement a kernel-based machine. In the specific case of an
RBF kernel of Gaussian style, the discriminant function

fx , represented by ( )

Equation (10.23), takes the form

2

xx

N

i

¦

fx () D e

2

b , (10.25)

2 V

i

i 0

which can be implemented as shown in Figure 10.5.

RBF
Center at x 1

D

x i1

y

D

Summ
-ation

RBF
Center at x 2

x i 2

sgn

:
:

:
:

:
:

D

x iN

RBF
Center at x N

Figure 10.5. Architecture of an RBF-based support vector machine

This implementation effectively represents the structure of an RBF-based
support vector machine in which the Gaussian activation functions are centred at
sampled values, and the multipliers

D play the role of interconnecting weights.

10.2.3 Applications

In engineering, support vector machines have found useful applications in
nonlinear regression estimation and in time series forecasting and prediction.
Nonlinear regression estimation addresses the problem of estimating a
function given by a set of data ( x , y ) , i = 1, 2, …, N , generated by an unknown

i i

function to be estimated, where

x are the sampled values of data set and

y are

di

the desired values to be estimated using the approximating function

N

¦

f (, ) xa a I () x b .

ii

i 0

In the above function, the functions I

i x

( ) are called features and

a are

coefficients to be estimated from given data by minimizing the functional