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Difference between revisions of "Auto-regression"

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A regressive dependence of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139701.png" /> of a given random sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139702.png" /> on the preceding values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139703.png" />. A linear auto-regression scheme of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139704.png" /> is defined by a linear [[Regression|regression]] equation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139707.png" />, i.e.
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139708.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A regressive dependence of the values of $X_n$ 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|regression]] equation between $X_n$ and $X_{n-k}$, $k=1,\ldots,m$, i.e.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a0139709.png" /> are constants and the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a01397010.png" /> are identically distributed with average zero, variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013970/a01397011.png" /> 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|time series]] (the concept of a linear auto-regression scheme was first introduced by [[Yule, George Udny|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-regressive process]].
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\begin{equation}
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\tag{*}
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X_n = \beta_1 X_{n-1} + \dots + \beta_m X_{n-m} + \epsilon_n ,
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\end{equation}
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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|time series]] (the concept of a linear auto-regression scheme was first introduced by [[Yule, George Udny|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-regressive process]].

Latest revision as of 01:22, 15 February 2024


A regressive dependence of the values of $X_n$ 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=52903
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article