State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 331

2

x x

i

RBF kernels Kxx (, ) e

2

2 V

i

sigmoid kernels Kxx ( , ) tanh[ ( cx x T

T

, ) ].

i i

The condition for a selected kernel to be acceptable as an inner product kernel
and to be useful for building a support vector machine is defined by Mercer's
theorem , which states that the proposed kernel function must be a symmetric
function, as defined by Equation (10.17). Furthermore, an inner product kernel to
be used in building the basic architecture of the support vector machine shown in
Figure 10.3 must be expandable in the series

f

¦

Kxx (, ) O k xk x () ( ) , (10.19)

i i i i i

i 1

where

O are eigenvalues and i kx are the eigenfunctions of the expansion.

( )

10.2.1 Data-dependent Representation

Auflauf and Biehl (1989), using a data-dependent representation , have worked out
a simple and fast convergent sequential algorithm for finding the optimal
parameters of a discriminant function with the largest margin. The algorithm that
they called adatron considers the discriminant function in terms of

N

¦

fx () sgn( D xx b

T

) , (10.20)

i i

i 0

where N is the number of samples and Dthe multipliers of individual samples that
should be selected so that the quadratic form

N

1

N N

¦ ¦

J () D D DD dd x x , , (10.21)

i i j i j i j

2

i 1 i 1 j 1

is optimized subject to the constraint

N

¦

D 0 , (10.22)

ii

i 1

for

D t , i = 1, 2, …, N , where .,. represents the inner product of

0

x and x .