Autocovariance

In statistics, given a real stochastic process X(t), the autocovariance is the covariance of the variable with itself, i.e. the variance of the variable against a time-shifted version of itself. If the process has the mean E[X_t] = μ_t, then the autocovariance is given by

$C_{XX}(t,s) = E[(X_t - \mu_t)(X_s - \mu_s)] = E[X_t X_s] - \mu_t \mu_s.\,$

where E is the expectation operator.

Stationarity

If X(t) is stationary process, then the following conditions are true:

$\mu_t = \mu_s = \mu \,$ for all t, s

and

$C_{XX}(t,s) = C_{XX}(s-t) = C_{XX}(\tau)\,$

where

$\tau = s - t\,$

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

As a result, the autocovariance becomes

$C_{XX}(\tau) = E[(X(t) - \mu)(X(t+\tau) - \mu)]\,$

$= E[X(t) X(t+\tau)] - \mu^2\,$

$= R_{XX}(\tau) - \mu^2,\,$

where R_XX represents the autocorrelation in the signal processing sense.

Normalization

When normalized by dividing by the variance σ², the autocovariance C becomes the autocorrelation coefficient function c,

$c_{XX}(\tau) = \frac{C_{XX}(\tau)}{\sigma^2}.\,$

The autocovariance function is itself a version of the autocorrelation function with the mean level removed. If the signal has a mean of 0, the autocovariance and autocorrelation functions are identical .

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 normalisation with the variance will put this into the range [−1, 1].

Properties

The autocovariance of a linearly filtered process $Y_t$

$Y_t = \sum_{k=-\infty}^\infty a_k X_{t+k}\,$

is $C_{YY}(\tau) = \sum_{k,l=-\infty}^\infty a_k a^*_l C_{XX}(\tau+k-l).\,$

See also

• Autocorrelation

References

• P. G. Hoel, Mathematical Statistics, Wiley, New York, 1984.

• Lecture notes on autocovariance from WHOI