State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 330

N

¦ w O y k x () , (10.14)

ii i

i 1

where the transformed value kx represents the feature vector corresponding to ( )

i

the input vector x .
After replacing the last Equation by equation (10.7), the separating surface in
the feature space is found as

N

¦

O yk

T

()( ) 0 x k x , (10.15)

ii i

i 1

or as

N

¦

O yK x x (, ) 0 , (10.16)

ii i

i 1

where

K xx (, ) kxkx kxkx

T

( )()

T

()( ) (10.17)

i i i

is the inner product kernel , which is a symmetric function, i.e.

Kxx (, ) Kx x ( , ) . (10.18)

i i

K ( x,x 1 )

x 1

y

x 2 Summ-

ation

K ( x,x 2 )

:
:

:
:

b

x N

K ( x,x N )

Figure 10.3. Basic architecture of a support vector machine

In practice, kernels of various shapes have been used:

T n

polynomial kernels Kxx ( , ) [( x x , ) 1]

i i