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 , then the autocovariance is given by
where E is the expectation operator.
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
- P. G. Hoel, Mathematical Statistics, Wiley, New York, 1984.
- Lecture notes on autocovariance from WHOI