State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 329

wkx .

() 0 (10.7)

Equation (10.6), as shown in Figure 10.2, can be used directly for implementation
of a kernel-based machine (Principe et al. , 1999).
In the following, we will seek for the optimal separating hyperplane (Haykin,
1999) using the set of training data samples (

x , y ) and the constraint

i i

T

t , i = 1, 2, …, N . This is achieved by optimal selection of the value
of b and by determination of the optimal value of w by minimizing the cost
function

ywx b ( ) 1

i i

1

T
J www

() . (10.8)

2

Using for this purpose the method of Lagrange multipliers, we have to minimize
the Lagrangian function

1

N

T T

Jwb (,,) O ww ¦ O [ ( y wx b ) 1] (10.9)

i i i

2

i 1

with respect to w and b and to maximize with respect to Ȝ by solving the equations

w Jwb

b

(,,) O

0 (10.10)

w

and

w Jwb

w

(,,) O

0 . (10.11)

w

As a result, the values of the weight vector w are found as

N

w ¦ O y x (10.12)

iii

i 1

under the condition that

N

O 0 , (10.13)

¦

ii

i 1

x , i.e .

holds. Taking into consideration the nonlinearly transformed value of

kx , the optimal value of w found above becomes

()
i