|Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.|
|See also: model finding, establishing ARIMA models|
Time Series - Definition of ARIMA Models
ARIMA (auto-regressive integrated moving average) models establish a powerful class of models which can be applied to many real time series. ARIMA models are based on three parts: (1) an autoregressive part, (2) a contribution from a moving average, and (3) a part involving the first derivative of the time series:
The auto-regressive part (AR) of the model has its origin in the theory that individual values of time series can be described by linear models based on preceding observations. For instance: x(t) = 3 x(t-1) - 4 x(t-2). The general formula for describing AR[p]-models (auto-regressive models) is:
The general description of MA[q]-models is:
When combining both AR and MA models, ARMA models are obtained. In general, forecasting with an ARMA[p,q]-model is described using the following equation:
After additional differentiation of the
time series, and integrating it after application of the model,
one speaks of ARIMA models. They are used when trend filtering is required.
The parameter d of the ARIMA[p,d,q]-model determines the number of differentiation
Then, a suitable ARMA[p,q] model is fitted to the resulting series.
Finally, the estimated forecasts have to be integrated d times.
Last Update: 2012-10-08