ARIMA by George Box and Gwilym Jenkins
George Box and Gwilym Jenkins, the statisticians named the time series analysis as Box–Jenkins model which implies “autoregressive moving average” (ARMA) or also known as ARIMA models in order to establish the perfect match for the time series.
Stationarity and seasonality – so as to develop a Box–Jenkins methodology, it is essential to identify that the time series is stationary or not and also if there exist any important seasonality which needs modification. Stationary can be determined from the run sequence plot. Run sequence plot is meant to showcase consistent and constant location and scale. However, stationarity can also be assessed with the help of autocorrelation plot but basically, we use autocorrelation plot for non stationarity with slow decay.
Seasonality – perodicity is the other name for seasonality and can be detected with the help of autocorrelation plot or from seasonal subseries plot or spectral plot.
• Box and Jenkins made use of differencing approach in order to attain stationarity. Also, fitting curve and difference of fitted values from original relevant data can be used for Box–Jenkins methodology.
• Identify p and q
• After identifying the stationarity and seasonality, it is necessary to determine the order of p and q of autoregressive and the moving average terms.
• Autocorrelation and incomplete autocorrelation plots
• The identification of the order is done with the help of autocorrelation plot and partial autocorrelation plot. Both, these plots are compared with theoretical behavior of these very plots to determine the order.
• The sample autocorrelation function must possess an exponentially decreasing appearance for the AR process particularly.
• In MA process, for lag q+1, the autocorrelation function is zero therefore, it is necessary to analyze the sample autocorrelation function to find where it grows zero.
The ARIMA Box-Jenkins model is the concatenation of the AR and MA models :
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