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((Correlation and covariance))
((See also|Cross-correlation))
((refimprove|date=December 2016))

In [[probability]] and [[statistics]], given two [[stochastic processes]] <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math>, the '''cross-covariance''' is a function that gives the [[covariance]] of one process with the other at pairs of time points. With the usual notation <math>\operatorname E</math> for the [[expected value|expectation]] [[Operator (mathematics)|operator]], if the processes have the [[mean]] functions <math>\mu_X(t) = \operatorname \operatorname E[X_t]</math> and <math>\mu_Y(t) = \operatorname E[Y_t]</math>, then the cross-covariance is given by

:<math>\operatorname{K}_{XY}(t_1,t_2) = \operatorname{cov} (X_{t_1}, Y_{t_2}) = \operatorname{E}[(X_{t_1} - \mu_X(t_1))(Y_{t_2} - \mu_Y(t_2))] = \operatorname{E}[X_{t_1} Y_{t_2}] - \mu_X(t_1) \mu_Y(t_2).\,</math>

Cross-covariance is related to the more commonly used [[cross-correlation]] of the processes in question.

In the case of two random vectors <math>\mathbf{X}=(X_1, X_2, \ldots , X_p)^{\rm T}</math> and <math>\mathbf{Y}=(Y_1, Y_2, \ldots , Y_q)^{\rm T}</math>, the cross-covariance would be a <math>p \times q</math> matrix <math>\operatorname{K}_{XY}</math> (often denoted <math>\operatorname{cov}(X,Y)</math>) with entries <math>\operatorname{K}_{XY}(j,k) = \operatorname{cov}(X_j, Y_k).\,</math> Thus the term ''cross-covariance'' is used in order to distinguish this concept from the covariance of a random vector <math>\mathbf{X}</math>, which is understood to be the [[covariance matrix|matrix of covariances]] between the scalar components of <math>\mathbf{X}</math> itself.

In [[signal processing]], the cross-covariance is often called [[cross-correlation]] and is a [[Similarity measure|measure of similarity]] of two [[signal (information theory)|signal]]s, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative [[time]] between the signals, is sometimes called the ''sliding [[dot product]]'', and has applications in [[pattern recognition]] and [[cryptanalysis]].

==Cross-covariance of random vectors==
((main|Cross-covariance matrix))

==Cross-covariance of stochastic processes==
The definition of cross-covariance of random vectors may be generalized to [[stochastic process|stochastic processes]] as follows:
===Definition===
Let <math>\{ X(t) \}</math> and <math>\{ Y(t) \}</math> denote stochastic processes. Then the cross-covariance function of the processes <math>K_{XY}</math> is defined by:<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.172))

((Equation box 1
|indent = :
|title=
|equation = ((NumBlk||<math>\operatorname{K}_{XY}(t_1,t_2) \stackrel{\mathrm{def)){=}\  \operatorname{cov} (X_{t_1}, Y_{t_2}) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math>|((EquationRef|Eq.1))))
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA))

where <math>\mu_X(t) = \operatorname{E}\left[X(t)\right]</math> and <math>\mu_Y(t) = \operatorname{E}\left[Y(t)\right]</math>.

If the processes are [[complex-valued]] stochastic processes, the second factor needs to be [[complex conjugate]]d:

:<math>\operatorname{K}_{XY}(t_1,t_2) \stackrel{\mathrm{def)){=}\  \operatorname{cov} (X_{t_1}, Y_{t_2}) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \overline{\left( Y(t_2)- \mu_Y(t_2) \right)} \right]</math>

===Definition for jointly WSS processes===
If <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are a [[Joint wide-sense stationarity|jointly wide-sense stationary]], then the following are true:

:<math>\mu_X(t_1) = \mu_X(t_2) \triangleq \mu_X</math> for all <math>t_1,t_2</math>,

:<math>\mu_Y(t_1) = \mu_Y(t_2) \triangleq \mu_Y</math> for all <math>t_1,t_2</math>

and

:<math>\operatorname{K}_{XY}(t_1,t_2) = \operatorname{K}_{XY}(t_2 - t_1,0)</math> for all <math>t_1,t_2</math>

By setting <math>\tau = t_2 - t_1</math> (the time lag, or the amount of time by which the signal has been shifted), we may define

:<math>\operatorname{K}_{XY}(\tau) =  \operatorname{K}_{XY}(t_2 - t_1) \triangleq \operatorname{K}_{XY}(t_1,t_2)</math>.

The cross-covariance function of two jointly WSS processes is therefore given by:

((Equation box 1
|indent = :
|title=
|equation = ((NumBlk||<math>\operatorname{K}_{XY}(\tau) = \operatorname{cov} (X_{t}, Y_{t-\tau}) = \operatorname{E}[(X_t - \mu_X)(Y_{t- \tau} - \mu_Y)] = \operatorname{E}[X_t Y_{t-\tau}] - \mu_X \mu_Y</math>|((EquationRef|Eq.2))))
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA))

which is equivalent to

:<math>\operatorname{K}_{XY}(\tau) = \operatorname{cov} (X_{t+\tau}, Y_{t}) = \operatorname{E}[(X_{t+ \tau} - \mu_X)(Y_{t} - \mu_Y)] = \operatorname{E}[X_{t+\tau} Y_t] - \mu_X \mu_Y</math>.

===Uncorrelatedness===
Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called '''uncorrelated''' if their covariance <math>\operatorname{K}_{\mathbf{X}\mathbf{Y))(t_1,t_2)</math> is zero for all times.<ref name=KunIlPark/>((rp|p.142)) Formally:

:<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y))(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.

==Cross-covariance of deterministic signals==
The cross-covariance is also relevant in [[signal processing]] where the cross-covariance between two [[wide-sense stationary]] [[random 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]] of one of the signals). For a large number of samples, the average converges to the true covariance.

Cross-covariance may also refer to a '''"deterministic" cross-covariance''' between two signals.  This consists of summing over ''all'' time indices. For example, for [[discrete-time]] signals <math>f[k]</math> and <math>g[k]</math> the cross-covariance is defined as

:<math>(f\star g)[n] \ \stackrel{\mathrm{def)){=}\  \sum_{k\in \mathbb{Z)) \overline{f[k]} g[n+k] = \sum_{k\in \mathbb{Z)) \overline{f[k-n]} g[k]</math>

where the line indicates that the [[complex conjugate]] is taken when the signals are [[complex-valued]].

For [[Continuous function|continuous functions]] <math>f(x)</math> and <math>g(x)</math> the (deterministic) cross-covariance is defined as

:<math>(f\star g)(x) \ \stackrel{\mathrm{def)){=}\  \int \overline{f(t)} g(x+t)\,dt = \int \overline{f(t-x)} g(t)\,dt</math>.

=== Properties ===

The (deterministic) cross-covariance of two continuous signals is related to the [[convolution]] by

:<math>(f \star g)(t) = (\overline{f(-\tau)}*g(\tau))(t)</math>

and the (deterministic) cross-covariance of two discrete-time signals is related to the [[discrete convolution]] by

:<math>(f \star g)[n] = (\overline{f[-k]}*g[k])[n]</math>.

==See also==
* [[Autocovariance]]
* [[Autocorrelation]]
* [[Correlation]]
* [[Convolution]]
* [[Cross-correlation]]

==References==
((reflist))

==External links==
* [http://mathworld.wolfram.com/Cross-Correlation.html Cross Correlation from Mathworld]
* http://scribblethink.org/Work/nvisionInterface/nip.html
* http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
* http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf

[[Category:Covariance and correlation]]
[[Category:Time domain analysis]]
[[Category:Signal processing]]

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