Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 60

CI 2RMSE nn ( p )

is preferred, where p represents the number of parameters estimated in the
forecasting method.

2.9.5.3 Forecasting Using Adaptive Smoothing
In adaptive smoothing , which is a more advanced version of exponential
smoothing, the smoothing constant D is adjusted on-line according to the actual
value of the forecast error. This is presented below on the example of the k -step
prediction equation

1

Cz
A z

()
()

y ( t + k ) = H( t +k ) (2.7)

1

with the polynomial operators

m

az

j

A = ( ) , with a 0

¦

j 0

jo

n

cz

i

C = , with 0 c 0

¦

i

i 0

and H( t ) as a white noise. The prediction is qualified as good if it minimizes the
cost function

{(

k VEyt

k }

Introducing the output prediction error

yt k ( ) yt k ( ) yt kt ˆ ( ) ,

where ˆ( yt kt ) is the predicted value

yt kt ˆ ( ) E yt { ( k t ) }

- ½

ª

1

º

° Cz () ( °

= E H tk t )

® ¾

« »

1

Az ()

¬ ¼

° °

¯ ¿

of the real future value y ( t + k ) using the minimum mean square error MMSE.
Introducing now the Diophantine equation

1 1 1 k 1

Cz () ()() Az Fz z Gz () (2.8)