# Auto-regression

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: http://encyclopediaofmath.org/index.php?title=Auto-regression&oldid=12046