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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.
 
Autocovariance is related to the more commonly used [[autocorrelation]] by the [[variance]] of the variable in question.
 
== Stationarity ==
 
If ''X''(''t'') is [[stationary process]], then the following are true:
 
:<math>\mu_t = \mu_s = \mu \,</math> for all ''t'', ''s''
 
and
 
:<math>C_{XX}(t,s) = C_{XX}(s-t) = C_{XX}(\tau)\,</math>
 
where
 
:<math>\tau = s - t\,</math>
 
is the lag time, or the amount of time by which the signal has been shifted.
 
As a result, the autocovariance becomes
 
:<math>C_{XX}(\tau) = E[(X(t) - \mu)(X(t+\tau) - \mu)]\,</math>
 
::::<math> = E[X(t) X(t+\tau)] - \mu^2\,</math>
 
::::<math> = R_{XX}(\tau) - \mu^2,\,</math>
 
== Normalization ==
 
When normalized by dividing by the [[variance]] &sigma;<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>
 
:<math>c_{XX}(\tau) = \frac{C_{XX}(\tau)}{\sigma^2}.\,</math>
However, often the autocovariance is called autocorrelation even if this normalization has not been performed.
 
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 &sigma;<sup>2</sup> indicating perfect correlation at that lag. The normalization with the variance will put this into the range [&minus;1,&nbsp;1].
 
== Properties ==
The autocovariance of a linearly filtered process <math>Y_t</math>
:<math>Y_t = \sum_{k=-\infty}^\infty a_k X_{t+k}\,</math>
:is <math>C_{YY}(\tau) = \sum_{k,l=-\infty}^\infty a_k a^*_l C_{XX}(\tau+k-l).\,</math>
 
== See also ==
* [[Autocorrelation]]
 
== References ==
 
* P. G. Hoel, Mathematical Statistics, Wiley, New York, 1984.
* [http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html Lecture notes on autocovariance from WHOI]
 
<references />
 
[[Category:Covariance and correlation]]
[[Category:Time series analysis]]
[[Category:Fourier analysis]]

Revision as of 23:35, 3 March 2014

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