Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 52

the one-step prediction

p

x ˆ a X P .

¦

t 1 i t t 1

i 1

2.9.4.2 Forecasting Using a Moving-average Model MA(q)
For the moving average model

q

x E e P e

¦

t i t i t

i 1

the estimated optimal linear forecast of j steps ahead is given by

q

ˆ

x ¦ E P

tj itij

i 1

2.9.4.3 Forecasting Using an ARMA Model
The general form of an ARMA process is written as

p q

yt ( ) ¦ D yt i ( ) ¦ E yt i ( ) Zt ( ) ,

i i

i 1 i 1

with Z ( t ) as white noise. For simplicity, the transcription

I ()() Byt T ()() BZt ,

is preferred, where

T are the corresponding polynomials.
Considering now the general form of a linear process

I () B

and

()

f

yt () T Zt i ( ) (2.1)

¦

i

i 0

the estimated forecast can be built as

t 1

yt k ˆ( ) wyt i ( ) . (2.2)

¦

i

i 0

Combining the last two equations we get the estimated forecast as

t 1 f

ˆ (

yt k ) w T Zt i ( j ).

¦¦

i j

i 0 j 0