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| In [[statistics]], given a real [[stochastic process]] ''X''(''t''), the '''autocovariance''' is the [[covariance]] of the variable against a time-shifted version of itself. If the process has the [[mean]] <math>E[X_t] = \mu_t</math>, then the autocovariance is given by
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| :<math>C_{XX}(t,s) = E[(X_t - \mu_t)(X_s - \mu_s)] = E[X_t X_s] - \mu_t \mu_s.\,</math>
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| where ''E'' is the [[expected value|expectation]] operator.
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| Autocovariance is related to the more commonly used [[autocorrelation]] by the [[variance]] of the variable in question.
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| == Stationarity ==
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| If ''X''(''t'') is [[stationary process]], then the following are true:
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| :<math>\mu_t = \mu_s = \mu \,</math> for all ''t'', ''s''
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| and | |
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| :<math>C_{XX}(t,s) = C_{XX}(s-t) = C_{XX}(\tau)\,</math>
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| where
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| :<math>\tau = s - t\,</math>
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| is the lag time, or the amount of time by which the signal has been shifted. | |
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| As a result, the autocovariance becomes
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| :<math>C_{XX}(\tau) = E[(X(t) - \mu)(X(t+\tau) - \mu)]\,</math>
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| ::::<math> = E[X(t) X(t+\tau)] - \mu^2\,</math>
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| ::::<math> = R_{XX}(\tau) - \mu^2,\,</math>
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| == Normalization ==
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| When normalized by dividing by the [[variance]] σ<sup>2</sup>, the autocovariance ''C'' becomes the [[autocorrelation]] ''coefficient'' function ''c'',<ref name="nonlinSystems">{{cite book|last=Westwick|first=David T.|title=Identification of Nonlinear Physiological Systems|year=2003|publisher=IEEE Press|isbn=0-471-27456-9|pages=17–18}}</ref>
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| :<math>c_{XX}(\tau) = \frac{C_{XX}(\tau)}{\sigma^2}.\,</math>
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| However, often the autocovariance is called autocorrelation even if this normalization has not been performed.
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| The autocovariance can be thought of as a measure of how similar a signal is to a time-shifted version of itself with an autocovariance of σ<sup>2</sup> indicating perfect correlation at that lag. The normalization with the variance will put this into the range [−1, 1].
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| == Properties ==
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| The autocovariance of a linearly filtered process <math>Y_t</math>
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| :<math>Y_t = \sum_{k=-\infty}^\infty a_k X_{t+k}\,</math>
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| :is <math>C_{YY}(\tau) = \sum_{k,l=-\infty}^\infty a_k a^*_l C_{XX}(\tau+k-l).\,</math>
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| == See also ==
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| * [[Autocorrelation]]
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| == References ==
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| * P. G. Hoel, Mathematical Statistics, Wiley, New York, 1984.
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| * [http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html Lecture notes on autocovariance from WHOI]
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| <references />
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| [[Category:Covariance and correlation]]
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| [[Category:Time series analysis]]
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| [[Category:Fourier analysis]]
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