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In statistics given a time series or continuous signal Xt then the autocovariance is simply the covariance of the signal against a time shifted version of itself. If the series has a constant mean, E[X_t] = μ then this is given by
- <math>\, \gamma(i,j) = E[(X_i - \mu)(X_j - \mu)]<math>.
Where E is the expectation of the function. If Xt is second order stationary then this definition becomes the more familiar
- <math>\, \gamma(k) = E[(X_i - \mu)(X_{i+k} - \mu)]<math>.
The k is the amount the signal has been shifted and is usually referred to as the lag. When normalised by dividing by the variance σ2 then the autocovariance becomes the autocorrelation R(k). That is
- <math> R(k) = \frac{\gamma(k)}{\sigma^2}<math>.
Note, however, that some disciplines use the terms autocovariance and autocorrelation interchangably.
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 normalisation with the variance will put this into the range [-1, 1].
References
- P. G. Hoel (1984): Mathematical Statistics, New York, Wiley
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