Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 54

2.9.4.4 Forecasting Using an ARIMA Model
The forecasting approaches presented so far refer only to stationary models. In
practice, however, many important time series are not stationary, so that they have
to be transformed to stationary time series. For instance, the generalization of an
ARMA model can be modified to provide a model for a time series that is
nonstationary in the mean (see Section 2.4.4). The modified version of an ARMA
is known as ARIMA ( i.e. the autoregressive integrated moving average ). The term
integrated indicates the fact that the model is produced by repeated integrating or
summing of the ARMA process. For example, by multiple summing the ARMA
process we get the ARIMA model

p q

y a y PE Z

¦ ¦

n i n i j n j

i 1 j 1

for

n t , where 0

n

x ¦ y .

n i

i 1

Using the last equation we can build

x x y

n n 1 n

which, after applying the z -transformation, results in

yz () (1 z

1

)() xz

,

so that the z -transformed ARIMA model is

1

az ()(1 z )() xz E () () zZz . P

Again, after d successive integrations, the last equation is converted to

az ()(1 z

1

) ()

d

xz E () () zZ z

,

P

This is the ARIMA( p , d , q ) model with p and q as the degrees of polynomials a ( z )
and ȕ ( z ) respectively.
We now consider the ARMA value

f

y t () T Z t ( i ) (2.5)

¦

i

i 0

and the prediction