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A regressive dependence of the values of of a given random sequence on the preceding values of . A linear auto-regression scheme of order is defined by a linear regression equation between and , , i.e.


where are constants and the random variables are identically distributed with average zero, variance and are uncorrelated (sometimes they are assumed to be independent). An auto-regression scheme is a useful stochastic model for the description of certain time series (the concept of a linear auto-regression scheme was first introduced by G. Yule in 1921) in order to analyze time series describing a system which is oscillating under the effect of internal forces and random external shocks. The auto-regression scheme (*) may be regarded as a stochastic process of a special type: an auto-regressive process.

How to Cite This Entry:
Auto-regression. A.V. Prokhorov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098