Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 55

f

yt k ˆ( ¦ ) T (2.6)

i t ki

ik

and build the prediction error

f f

ˆ

yt () ¦ T Zt i ( ) yt k ( ) ¦ T Z ,

i i t k i

i 0 i k

or definitely

k 1

yt () ¦ E .

i t ki

i 0

2.9.4.5 Forecasting Using a CARIMAX Model
The predictive capability of the CARIMAX model is discussed in detail, along
with its application to predictive control, in Section 2.10.6.

2.9.5 Forecasting Using Smoothing

Processing of sampled signals mainly includes

x signal smoothing , i.e . optimal estimation of a signal value within a given
time interval, based on signal values within the interval
x signal filtering , i.e. optimal estimation of actual signal value at the present
point based on the past and the present sampled values of the signal
x signal prediction , i.e. optimal estimation of future signal values based on
the past and the present sampled values

In time series analysis, smoothing is a technique focused on reduction of
irregularities or random fluctuations in time series data in order to provide a clean
time series pattern out of contaminated observation data. The simplest smoothing
technique used is moving-average smoothing , as well as its more advanced
modification, i.e. exponential smoothing .

2.9.5.1 Forecasting Using a Simple Moving Average
Moving averages are used for prediction of future values based on weighted
averages of the past values. They are useful in reducing the random variations
present in observation data. For example, the moving average that uses n past
observations and the most recent one to calculate the next time series value is

x t ( ) xt ( 1) ... xt ( n )

xt ( ) .

m

n

Some modifications of the moving average are