Neural Networks Approach - Computational Intelligence in Time Series Forecasting - page 124

( GI wy O ) .

Here, I represents the n -dimensional identity matrix and G is the corresponding
Green's matrix

ª Gx x (,) (, ) . (, )

(,) (,) .(,)
... ...
(,) (,) .(,)

Gx x Gx x º

11 1 2 1 n

« »

Gx x Gx x Gx x

« »

21 21 2 n

G « ,

»

« »

Gx x Gx x Gx x

« »

¬ ¼

n 1 n 2 n n

which is a symmetric matrix with the property

Gx x (, ) Gx x ( , )

i j j i

because the identity matrix I is also symmetric.
From the solution equation

n

f () x ¦ Gx x (, )

i i

i 1

the corresponding regularization network (Figure 3.18) can be structured. The
input layer of the network has an equivalent number of units to the dimension of
the input vector, i.e . to the number of independent variables of the problem to be
solved. The subsequent hidden layer, fully connected with the input layer with the
fixed value weights, has the same number of nonlinear units as the number of data
points and the activation function in the form of a Green's function with the output
(, ).
i

Gxx It does not participate in the training process. Finally, the output layer,
also fully connected to the hidden layer, contains one or more linear units with the
weights w that correspond to the unknown coefficients of the above solution
equation.

G

w 1

x 1

w 2

G

x 2

f(x)

:
:

:
:

:
:

:
:

w n

x n

G

Figure 3.18. Regularization network