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Auto-regression

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A regressive dependence of the values of of a given random sequence \{X_n : n=0, \pm1, \ldots\} on the preceding values of X_{n-1}, \ldots, X_{n-m}. A linear auto-regression scheme of order m is defined by a linear regression equation between X_n and X_{n-k}, k=1,\ldots,m, i.e.

\begin{equation} \tag{*} X_n = \beta_1 X_{n-1} + \dots + \beta_m X_{n-m} + \epsilon_n , \end{equation}

where \beta_1, \ldots, \beta_m are constants and the random variables \epsilon_n are identically distributed with average zero, variance \sigma^2 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Auto-regression&oldid=55492
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article