Given a time series of data where is an integer index and the are real numbers, then an ARMA(p,q) model is given by
An ARIMA(p,d,q) process is obtained by integrating an ARMA(p,q) process. That is,
It should be noted that not all choices of parameters produce well-behaved models. In particular, if the model is required to be stationary then conditions on these parameters must be met.
Some well-known special cases arise naturally. For example, an ARIMA(0,1,0) model is given by:
A number of variations on the ARIMA model are commonly used. For example, if multiple time series are used then the can be thought of as vectors and a VARIMA model may be appropriate. Sometimes a seasonal effect is suspected in the model. For example, consider a model of daily road traffic volumes. Weekends clearly exhibit different behaviour from weekdays. In this case it is often considered better to use a SARIMA (seasonal ARIMA) model than to increase the order of the AR or MA parts of the model. If the time-series is suspected to exhibit long-range dependence then the parameter may be replaced by certain non-integer values in a Fractional ARIMA (FARIMA also sometimes called ARFIMA) model.
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