Editing Autocorrelation (section)

=== Normalization ===
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 autocorrelation coefficient of a stochastic process is<ref name=KunIlPark/>{{rp|p.169}}

<math display=block>\rho_{XX}(t_1,t_2) = \frac{\operatorname{K}_{XX}(t_1,t_2)}{\sigma_{t_1}\sigma_{t_2}} = \frac{\operatorname{E}\left[(X_{t_1} - \mu_{t_1}) \overline{(X_{t_2} - \mu_{t_2})} \right]}{\sigma_{t_1}\sigma_{t_2}} .</math>

If the function <math>\rho_{XX}</math> is well defined, its value must lie in the range <math>[-1,1]</math>, with 1 indicating perfect correlation and −1 indicating perfect [[anti-correlation]].

For a [[Stationary process#wide-sense stationarity|wide-sense stationary]] (WSS) process, the definition is

<math display=block>\rho_{XX}(\tau) = \frac{\operatorname{K}_{XX}(\tau)}{\sigma^2} = \frac{\operatorname{E} \left[(X_{t+\tau} - \mu)\overline{(X_{t} - \mu)}\right]}{\sigma^2}</math>.

The normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of [[statistical dependence]], and because the normalization has an effect on the statistical properties of the estimated autocorrelations.