Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 27

ZZ I I Z ... I Z aa T T a ... T a

t 1 t 1 2 t 2 p t p t 1 t 1 2 t 2 q t q

and rearranging it as

2 p 2 q

(1 II BB ... I BZ ) (1 TT BB ... T Ba )

1 2 p t 1 q t

the model can finally be written in compact form as

I () BZ T () Ba ,

t t

where B is a delay operator. The derived compact model contains ( p+q +2)
unknown parameters TT TV that are to be estimated from

the given time series data. In practice, for the representation of actually occurring
stationary time series, it is frequently adequate enough to take p and q not greater
than 2. The presence of both autoregressive and moving-average terms in the
ARMA model enables the representation of complex time series with fewer
parameters than would be needed using a corresponding AR model.

PII I and

2

,, , p , ,..., ,

12 q a

12

2.5.4 ARIMA Model

This Box-Jenkins variant of the ARMA model is predestinated for applications to
nonstationary time series that become stationary after their differencing.
Differencing is an operation by which a new time series is built by taking the
successive differences of successive values, such as X ( t ) - X ( t- 1) along the
nonstationary time series pattern. In the acronym ARIMA, the letter I stands for
integrated .
The widely accepted convention for defining the structure of ARIMA models is
ARIMA( p , q , d ), where p stands for the number of autoregressive parameters, q is
the number of moving-average parameters, and d is the number of differencing
passes. For instance, the ARIMA(2, 3, 1) model has two autoregressive parameters,
three moving-average parameters, computed after the series have been differenced
once.
A variety of time series encountered in industry and business exhibit
nonstationary behaviour. In particular, they do not vary about a fixed mean
because of the possible presence of a drift component. Such time series may,
nevertheless, exhibit homogeneous behaviour of a kind. In particular, although the
general level about which fluctuations are occurring may be different at different
times, the broad behaviour of the series, when differences in level are allowed for,
may be similar. It can be shown that such behaviour may be represented by a
generalized autoregressive operator

,

d

M B I B 1 B