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In probability and statistics, given two stochastic processes { X t } {\displaystyle \left\{X_{t}\right\}} and { Y t } {\displaystyle \left\{Y_{t}\right\}} , 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 E {\displaystyle \operatorname {E} } for the expectation operator, if the processes have the mean functions μ X = E {\displaystyle \mu _{X}=\operatorname {\operatorname {E} } } and μ Y = E {\displaystyle \mu _{Y}=\operatorname {E} } , then the cross-covariance is given by
Cross-covariance is related to the more commonly used cross-correlation of the processes in question.
In the case of two random vectors X = T {\displaystyle \mathbf {X} =^{\rm {T}}} and Y = T {\displaystyle \mathbf {Y} =^{\rm {T}}} , the cross-covariance would be a p × q {\displaystyle p\times q} matrix K X Y {\displaystyle \operatorname {K} _{XY}} {\displaystyle \operatorname {cov} } ] with entries K X Y = cov . {\displaystyle \operatorname {K} _{XY}=\operatorname {cov}.\,} Thus the term cross-covariance is used in order to distinguish this concept from the covariance of a random vector X {\displaystyle \mathbf {X} } , which is understood to be the matrix of covariances between the scalar components of X {\displaystyle \mathbf {X} } itself.
In signal processing, the cross-covariance is often called cross-correlation and is a measure of similarity of two signals, 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.