Auto-Regressive Simulation (Second-Order)

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows realizations of a second-order auto-regressive (AR) 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 constants and are the auto-regressive constants. If , the series is stationary. 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 AR series of changes in the two constants. If the constants , do not meet the stationarity conditions, the series will diverge. The Demonstration is for a second-order process only. Additional AR terms would enable somewhat more complex series to be generated, but the differences from second-order processes would be difficult to ascertain.

For a detailed description of an AR process, see, for instance, G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications, San Francisco: Holden-Day, 1968 or G. Box, G. M. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed., Englewood Cliffs, NJ: Prentice-Hall, 1994.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.