Autocovariance

In probability and statistics, given a stochastic process $X=(X_{t})$, 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 $\mu _{t}=E[X_{t}]$, then the autocovariance is given by

$C_{XX}(t,s)=cov(X_{t},X_{s})=E[(X_{t}-\mu _{t})(X_{s}-\mu _{s})]=E[X_{t}X_{s}]-\mu _{t}\mu _{s}.\,$

Autocovariance is related to the more commonly used autocorrelation of the process in question.

In the case of a random vector $X=(X_{1},X_{2},...,X_{n})$, the autocovariance would be a square n by n matrix $C_{XX}$ with entries $C_{XX}(j,k)=cov(X_{j},X_{k}).\,$ This is commonly known as the covariance matrix or matrix of covariances of the given random vector.

Stationarity

If X(t) is stationary process, then the following 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

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

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

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

where $R_{XX}(\tau )$ is the autocorrelation of the signal with variance $\sigma ^{2}$. Some authors do not normalize the autocorrelation function. In those literatures, $C_{XX}(\tau )=R_{XX}(\tau )-\mu ^{2},\,$.

Normalization

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

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

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].

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
• Covariance and Correlation
• Covariance mapping
• Cross-covariance
• Cross-correlation
• Noise covariance estimation (as an application example)

References

• Hoel, P. G. (1984). Mathematical Statistics (Fifth ed.). New York: Wiley. ISBN 0-471-89045-6.
• Lecture notes on autocovariance from WHOI