Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 87

states that the expression for the MA( q ) [Eqn. (4)] can be rewritten (inverted)
into an autoregressive form (of infinite order) if the parameters of the moving
average model make the model 'invertible'. Note that the invertibility condition
of a moving average process is analogous to the stationarity condition of an
autoregressive process.

5.1.4 Autoregressive Moving Average (ARMA) Process

The autoregressive AR( p ) and moving average MA( q ) processes are special
cases of an autoregressive moving average process. An autoregressive moving
average (ARMA) process of order ( p , q ) denoted by ARMA( p , q ) represents a
real stochastic process x t with the following expression:

(5)

!

x PI x I x I x H T H T

t 1 t1 2 t 2 p t p t 1 t1 2

H TH !

t2 q tq

An ARMA( p ,0) process with p 1 is obviously an AR( p ) process, whereas
an ARMA(0, q ) process with q 1 is an MA( q ) process.

5.1.5 Autoregressive Integrated Moving Average (ARIMA)
Process

Stochastic modelling and forecasting of a time series requires adequate
knowledge about mathematical techniques for identifying patterns in time
series data and for expressing the physical process in terms of the mathematical
model. However, the physical processes are very complex in nature, the patterns
of time series data are unclear, and individual data points involve considerable
error. Hence, it is highly challenging in practice to explore the hidden patterns
in the data and also to generate forecasts. Box and Jenkins (1976) developed
an autoregressive integrated moving average (ARIMA) model and successfully
demonstrated their applications in forecasting of physical processes. The
ARIMA modelling is inherently a very powerful technique and contains great
flexibility. However, the ARIMA modelling requires a great deal of experience
because it is complex to understand and it is not easy to use. The ARIMA
modelling may often produce satisfactory results but the results entirely depend
on the analyst's/scientist's level of expertise (Bails and Peppers, 1982).
The general ARIMA model includes autoregressive as well as moving
average parameters, and explicitly includes differencing in the formulation of
the model (Box and Jenkins, 1976). Three specific parameters of a general
ARIMA model are: the autoregressive parameters ( p ), the number of
differencing passes ( d), and moving average parameters ( q ). Box and Jenkins
(1976) denoted the autoregressive integrated moving average process as
ARIMA ( p , d , q ), which means the ARIMA model contains ' p ' autoregressive
parameters and ' q ' moving average parameters which were computed for the
series after it was differenced ' d ' times.