Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 62

where

V is the variance of noise İ .
Finally, the minimum of

V is

1

Gz ()

yt kt ˆ( ) ( )

()

yt

1

Cz

Taking into account the k -step prediction defined by Equation (2.7) and the
Diophantine Equation (2.8), this result is now used to find the predictor value
ˆ(

from the relation

yt kt )

y ( t+k ) = yt kt ˆ( ) F z (

1

) ( et k , )

which is equivalent to

1

yt () F z ( ) () et .

This finally results in

1

Gz ()

yt kt ˆ( ) yt . ( )

1) 1

Az ( ) F z ( )

2.9.5.4 Combined Forecast
Thus far, various traditional methods available for time series forecasting have
been presented. It was mentioned that, unfortunately, there are no specific
guidelines for selection of a best forecasting method to solve a forecasting
problem. Besides, not each available method, applied to the same problem, delivers
the forecasting results with the same accuracy. For example, to forecast a
nonstationary, non-seasonal time series one can use the autoregressive method,
Holt-Winter's exponential smoothing technique, the Box-Jenkins ARMA/ARIMA
method, Kalman filtering, etc . Different methods will, for a given time series,
provide different forecasting results, so that, after comparing the individual
forecasting results, a decision has to be made about what prediction method should
be ultimately selected for further considerations. This is a difficult task requiring
much professional experience. As a way out of the selection dilemma the
nonlinear combination of forecasts has been advocated, as described below.
The need for combined forecast of a time series has been well understood for a
long time. Many studies have been done and revealed that not any arbitrary
combination of methods is decisive for an improved forecast, but it is essential that
the combination is nonlinear . Only the nonlinearity provides a combination with
better forecasts than either of the combination components separately, due to a
kind of synergic effect generated. It was also revealed that the forecasting results