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
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
- Covariance and Correlation
- Covariance mapping
- Noise covariance estimation (as an application example)
- Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.