Autoregressive Moving-Average Simulation (First Order)

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This Demonstration shows realizations of a first-order ARMA process , using the random variable drawn from a normal density with mean zero and variance unity. It is governed by the equation:


, where is the length of the series.

The constant is the autoregressive constant (), and the constant is the moving-average constant (). A series of length 400 is created in every case.


Contributed by: David von Seggern (University of Nevada) (March 2011)
Open content licensed under CC BY-NC-SA



The Demonstration is set such that the same random series of points is used no matter how the constants and are varied. However, when the "randomize" button is pressed, a new random series will be generated and used. Keeping the random series identical allows the user to see exactly the effects on the ARMA series of changes in the two constants. The constant is limited to (-1,1) because divergence of the ARMA series results when .

The Demonstration is for a first-order process only. Additional AR terms would enable more complex series to be generated, while additional MA terms would increase the smoothing.

For a detailed description of ARMA processes, see, for instance, G. Box, G. M. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed., Englewood Cliffs, NJ: Prentice-Hall, 1994.

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