In probability theory and statistics, given a stochastic process , the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. With the usual notation E for the expectation operator, if the process has the mean function , then the autocovariance is given by
Autocovariance is related to the more commonly used autocorrelation of the process in question.
In the case of a multivariate random vector , the autocovariance becomes a square n by n matrix, , with entry given by and commonly referred to as autocovariance matrix associated with vectors and .
If X(t) is a weakly stationary process, then the following are true:
- for all t, s
where is the lag time, or the amount of time by which the signal has been shifted.
When normalizing the autocovariance, C, of a weakly stationary process with its variance, , one obtains the autocorrelation coefficient :
The autocovariance of a linearly filtered process
- ^ Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.