Difference between revisions of "Autocovariance"
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| + | $#C+1 = 8 : ~/encyclopedia/old_files/data/A013/A.0103990 Autocovariance | ||
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| − | + | ''of a stochastic process $ X _ {t} $'' | |
| − | The term "autocovariance" is usually applied to stationary (in the wide sense) stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]). For such processes the autocovariance depends only on | + | The [[Covariance|covariance]] of $ X _ {t} $ |
| + | and $ X _ {t+h} $. | ||
| + | If $ {\mathsf E} X $ | ||
| + | denotes the mathematical expectation of a random variable $ X $, | ||
| + | then the autocovariance equals | ||
| + | |||
| + | $$ | ||
| + | {\mathsf E} ( X _ {t} - {\mathsf E} X _ {t} ) ( X _ {t+h} - {\mathsf E} | ||
| + | X _ {t+h} ). | ||
| + | $$ | ||
| + | |||
| + | The term "autocovariance" is usually applied to stationary (in the wide sense) stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]). For such processes the autocovariance depends only on $ h $ | ||
| + | and differs from the [[Auto-correlation|auto-correlation]] by the presence of a single factor which is equal to the variance of $ X _ {t} $. | ||
| + | The terms "covariance function" and "autocovariance function" are used together with the term "autocovariance" . | ||
Latest revision as of 18:48, 5 April 2020
of a stochastic process $ X _ {t} $
The covariance of $ X _ {t} $ and $ X _ {t+h} $. If $ {\mathsf E} X $ denotes the mathematical expectation of a random variable $ X $, then the autocovariance equals
$$ {\mathsf E} ( X _ {t} - {\mathsf E} X _ {t} ) ( X _ {t+h} - {\mathsf E} X _ {t+h} ). $$
The term "autocovariance" is usually applied to stationary (in the wide sense) stochastic processes (cf. Stationary stochastic process). For such processes the autocovariance depends only on $ h $ and differs from the auto-correlation by the presence of a single factor which is equal to the variance of $ X _ {t} $. The terms "covariance function" and "autocovariance function" are used together with the term "autocovariance" .
How to Cite This Entry:
Autocovariance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Autocovariance&oldid=13706
Autocovariance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Autocovariance&oldid=13706
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article