In probability 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.
If X(t) is stationary process, then the following are true:
- for all t, s
is the lag time, or the amount of time by which the signal has been shifted.
As a result, the autocovariance becomes
where is the autocorrelation of the signal.
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 σ2 indicating perfect correlation at that lag. The normalization with the variance will put this into the range [−1, 1].
The autocovariance of a linearly filtered process