Neural Networks Approach - Computational Intelligence in Time Series Forecasting - page 123

hand side of the equation, with the kernel defined by Green's function of the
differential operator

PP

ˆ

PPG x x (, ) G . ( x x )

i i

Bearing in mind the definition of Green's function and taking into account the
presence of the delta function on the right-hand side of the equation, the integral
transformation will generate a discrete sum of terms, so that the function f can be
defined as

1

n

fx () ¦ ( y fx Gxx ( )) (, )

,

i i i

O

i 1

where G ( x , x i ) is Green's function centred at x i . The last equation represents the
solution of the regularization problem as a linear combination of n Green's
functions with the expansion centre x i and expansion coefficients ( y i í f ( x i )).
Consequently, the solution of the regularization problem lies in the n -dimensional
subspace of the space of smooth functions, with the n Green's functions as its basis
(Poggio and Girosi, 1990). Furthermore, the basis function depends on stabilizer P ,
that represents the a priori knowledge of the problem domain as a kind of
constraint.
Introducing the definition of the expansion weights as

yf () x

w

i i

,

i

O

the above solution equation becomes

n

f () x ¦ Gx x (, ) .

i i

i 1

w , the last two equations have to be

Now, to determine the expansion weights

written in matrix form as

1 (

wy f )

O

and

f Gw

which result in