Editing Cross-correlation (section)

==Cross-correlation of stochastic processes==
In [[time series analysis]] and [[statistics]], the cross-correlation of a pair of [[random processes|random process]] is the correlation between values of the processes at different times, as a function of the two times. Let <math>(X_t, Y_t)</math> be a pair of random processes, and <math>t</math> be any point in time (<math>t</math> may be an [[integer]] for a [[discrete-time]] process or a [[real number]] for a [[continuous-time]] process). Then <math>X_t</math> is the value (or [[Realization (probability)|realization]]) produced by a given run of the process at time <math>t</math>.

=== Cross-correlation function ===
Suppose that the process has means <math>\mu_X(t)</math> and <math>\mu_Y(t)</math> and variances <math>\sigma_X^2(t)</math> and <math>\sigma_Y^2(t)</math> at time <math>t</math>, for each <math>t</math>. Then the definition of the cross-correlation between times <math>t_1</math> and <math>t_2</math> is<ref name=Gubner/>{{rp|p.392}}<math display="block">\operatorname{R}_{XY}(t_1, t_2) \triangleq\ \operatorname{E}\left[X_{t_1} \overline{Y_{t_2}}\right]</math>where <math>\operatorname{E}</math> is the [[expected value]] operator. Note that this expression may be not defined.

=== Cross-covariance function ===
Subtracting the mean before multiplication yields the cross-covariance between times <math>t_1</math> and <math>t_2</math>:<ref name=Gubner/>{{rp|p.392}}<math display="block">\operatorname{K}_{XY}(t_1, t_2) \triangleq\ \operatorname{E}\left[\left(X_{t_1} - \mu_X(t_1)\right)\overline{(Y_{t_2} - \mu_Y(t_2))}\right]</math>Note that this expression is not well-defined for all time series or processes, because the mean or variance may not exist.

===Definition for wide-sense stationary stochastic process===
Let <math>(X_t, Y_t)</math> represent a pair of [[stochastic process]]es that are [[Joint stationarity|jointly wide-sense stationary]]. Then the [[Cross-covariance of stochastic processes|cross-covariance function]] and the cross-correlation function are given as follows.

====Cross-correlation function====
<math display="block">\operatorname{R}_{XY}(\tau) \triangleq\ \operatorname{E}\left[X_t \overline{Y_{t+\tau}}\right]</math> or equivalently <math display="block">\operatorname{R}_{XY}(\tau) = \operatorname{E}\left[X_{t-\tau} \overline{Y_{t}}\right]</math>

==== Cross-covariance function====
<math display="block">\operatorname{K}_{XY}(\tau) \triangleq\ \operatorname{E}\left[\left(X_t - \mu_X\right)\overline{\left(Y_{t+\tau} - \mu_Y\right)}\right]</math> or equivalently <math display="block">\operatorname{K}_{XY}(\tau) = \operatorname{E}\left[\left(X_{t-\tau} - \mu_X\right)\overline{\left(Y_{t} - \mu_Y\right)}\right]</math>where <math>\mu_X</math> and <math>\sigma_X</math> are the mean and standard deviation of the process <math>(X_t)</math>, which are constant over time due to stationarity; and similarly for <math>(Y_t)</math>, respectively. <math>\operatorname{E}[\ ]</math> indicates the [[expected value]]. That the cross-covariance and cross-correlation are independent of <math>t</math> is precisely the additional information (beyond being individually wide-sense stationary) conveyed by the requirement that <math>(X_t, Y_t)</math> are ''jointly'' wide-sense stationary.

The cross-correlation of a pair of jointly [[wide sense stationary]] [[stochastic processes]] can be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a [[sampling (statistics)|sub-sampling]]{{which|date=May 2015}} of one of the signals). For a large number of samples, the average converges to the true cross-correlation.

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

The definition of the normalized cross-correlation of a stochastic process is<math display="block">
  \rho_{XX}(t_1, t_2) =
  \frac{\operatorname{K}_{XX}(t_1, t_2)}{\sigma_X(t_1)\sigma_X(t_2)} =
  \frac{\operatorname{E}\left[\left(X_{t_1} - \mu_{t_1}\right)\overline{\left(X_{t_2} - \mu_{t_2}\right)}\right]}{\sigma_X(t_1)\sigma_X(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 jointly wide-sense stationary stochastic processes, the definition is<math display="block">
  \rho_{XY}(\tau) =
  \frac{\operatorname{K}_{XY}(\tau)}{\sigma_X \sigma_Y} =
  \frac{\operatorname{E}\left[\left(X_t - \mu_X\right) \overline{\left(Y_{t+\tau} - \mu_Y\right)}\right]}{\sigma_X \sigma_Y}
</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.

===Properties===
====Symmetry property====
For jointly wide-sense stationary stochastic processes, the cross-correlation function has the following symmetry property:<ref name=KunIlPark>Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3</ref>{{rp|p.173}}<math display="block">\operatorname{R}_{XY}(t_1, t_2) = \overline{\operatorname{R}_{YX}(t_2, t_1)}</math>Respectively for jointly WSS processes:<math display="block">\operatorname{R}_{XY}(\tau) = \overline{\operatorname{R}_{YX}(-\tau)}</math>