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 (autoregressive 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 autoregressive 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(t1)  4 x(t2). The general formula for describing AR[p]models (autoregressive 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
steps.
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: 20121008