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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

where t and s are two time periods or moments in time.

Autocovariance is closely 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 the autocovariance matrix associated with vectors and .

Weak stationarity[edit]

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 :[1]

with .


The autocovariance of a linearly filtered process


See also[edit]


  1. ^ Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.