Autocovariance

In probability theory 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)={\text{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 closely related to the more commonly used autocorrelation of the process in question.

In the case of a multivariate random vector $X=(X_{1},X_{2},...,X_{n})$ , the autocovariance becomes a square n by n matrix, $C_{{XX}}$ , with entry $i,j$ given by $C_{X_{i}X_{j}}(t,s)={\text{cov}}(X_{i,t},X_{j,s})$ and commonly referred to as the autocovariance matrix associated with vectors $X_{t}$ and $X_{s}$ .

Weak stationarity

If X(t) is a weakly 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.

Normalization

When normalizing the autocovariance, C, of a weakly stationary process with its variance, $C_{XX}(0)=\sigma ^{2}$ , one obtains the autocorrelation coefficient $\rho$ :^[1]

\rho _{XX}(\tau )={\frac {C_{XX}(\tau )}{\sigma ^{2}}}

with $-1\leq \rho _{XX}(\tau )\leq 1$ .

Properties

The autocovariance of a linearly filtered process $Y_{t}$

Y_{t}=\sum _{{k=-\infty }}^{\infty }a_{k}X_{{t+k}}\,

C_{YY}(\tau )=\sum _{k,l=-\infty }^{\infty }a_{k}a_{l}C_{XX}(\tau +k-l).\,

References

↑ Westwick, David T. (2003). Identification of Nonlinear Physiological Systems. IEEE Press. pp. 17–18. ISBN 0-471-27456-9.

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

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Autocovariance

Weak stationarity

Normalization

Properties

See also

References