Auto-regression
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.
Auto-regression. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Auto-regression&oldid=55492