Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
v t e

In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question.

Auto-covariance of stochastic processes

Definition

With the usual notation ${\displaystyle \operatorname {E} }$ for the expectation operator, if the stochastic process ${\displaystyle \left\{X_{t}\right\}}$ has the mean function ${\displaystyle \mu _{t}=\operatorname {E} }$, then the autocovariance is given by^{[1]: p. 162}

${\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {cov} \left=\operatorname {E} =\operatorname {E} -\mu _{t_{1}}\mu _{t_{2}}}$

(Eq.1)

where ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ are two instances in time.

Definition for weakly stationary process

If ${\displaystyle \left\{X_{t}\right\}}$ is a weakly stationary (WSS) process, then the following are true:^{[1]: p. 163}

${\displaystyle \mu _{t_{1}}=\mu _{t_{2}}\triangleq \mu }$ for all ${\displaystyle t_{1},t_{2}}$

and

${\displaystyle \operatorname {E} <\infty }$ for all ${\displaystyle t}$

and

${\displaystyle \operatorname {K} _{XX}(t_{1},t_{2})=\operatorname {K} _{XX}(t_{2}-t_{1},0)\triangleq \operatorname {K} _{XX}(t_{2}-t_{1})=\operatorname {K} _{XX}(\tau ),}$

where ${\displaystyle \tau =t_{2}-t_{1}}$ is the lag time, or the amount of time by which the signal has been shifted.

The autocovariance function of a WSS process is therefore given by:^{[2]: p. 517}

${\displaystyle \operatorname {K} _{XX}(\tau )=\operatorname {E} (X_{t}-\mu _{t})(X_{t-\tau }-\mu _{t-\tau })=\operatorname {E} X_{t}X_{t-\tau }-\mu _{t}\mu _{t-\tau }}$

(Eq.2)

which is equivalent to

${\displaystyle \operatorname {K} _{XX}(\tau )=\operatorname {E} (X_{t+\tau }-\mu _{t+\tau })(X_{t}-\mu _{t})=\operatorname {E} X_{t+\tau }X_{t}-\mu ^{2}}$.

Normalizationedit

It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.

The definition of the normalized auto-correlation of a stochastic process is

${\displaystyle {\mathrm{\rho <!--\; \rho \; -->}}_{XX}(t_1,t_2)=\frac{{\mathrm{K}}_{XX}<!--\; \; -->(t_1,t_2}{}}$Zdroj:https://en.wikipedia.org?pojem=Autocovariance
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