Time series forecasting means the prediction of future values based upon a collection of present and past values. Accurate forecasting is of importance in many fields, as for instance for government activities, in the industry, which deal with predictions of future revenues, costs, demands, a.o. [123].

A time-series *X*(*t*) is defined as a series of values

where *t* is a discrete time variable.

Prediction for a time series can be defined as follows:

Given a time seriesX(t)and a moment in timet,there is to be determined the value of the series at the pointt+1,as a function of some of the previous values:

where

Tue Feb 9 20:20:27 JST 1999