Autocovariance

In statistics, given a real stochastic process "X"("t"), the autocovariance is simply the covariance of the signal against a time-shifted version of itself. If each state of the series has a mean, E ["X"_"t"] = μ_"t", then the autocovariance is given by

:$,\; K\_mathrm\{XX\}\; (t,s)\; =\; E\; [(X\_t\; -\; mu\_t)(X\_s\; -\; mu\_s)]\; =\; E\; [X\_tcdot\; X\_s]\; -mu\_tcdotmu\_s.,$

where E is the expectation operator.

Stationarity

If "X"("t") is wide sense stationary then the following conditions are true:

:$mu\_t\; =\; mu\_s\; =\; mu\; ,$ for all "t", "s"

and

:$K\_mathrm\{XX\}(t,s)\; =\; K\_mathrm\{XX\}(s-t)\; =\; K\_mathrm\{XX\}(\; au)\; ,$

where

:$au\; =\; s\; -\; t\; ,$

is the lag time, or the amount of time by which the signal has been shifted.

As a result, the autocovariance becomes

:$,\; K\_mathrm\{XX\}\; (\; au)\; =\; E\; \{\; (X(t)\; -\; mu)(X(t+\; au)\; -\; mu)\; \}$

::::$=\; E\; \{\; X(t)cdot\; X(t+\; au)\; \}\; -mu^2,,$

::::$=\; R\_mathrm\{XX\}(\; au)\; -\; mu^2,,$

where "R"_XX represents the autocorrelation, in the signal processing sense.

Normalization

When normalized by dividing by the variance σ² then the autocovariance becomes the autocorrelation coefficient ρ. That is

:$ho\_mathrm\{XX\}(\; au)\; =\; frac\{\; K\_mathrm\{XX\}(\; au)\}\{sigma^2\}.,$

Note, however, that some disciplines use the terms autocovariance and autocorrelation interchangeably.

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] .

See also

* Autocorrelation

References

* P. G. Hoel (1984): Mathematical Statistics, New York, Wiley

* [http://w3eos.whoi.edu/12.747/notes/lect06/l06s02.html Lecture notes on autocovariance from WHOI]