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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 ( t ) = E ⁡ [ X t ] {\displaystyle \mu _{X}(t)=\operatorname {\operatorname {E} } [X_{t}]} and μ Y ( t ) = E ⁡ [ Y t ] {\displaystyle \mu _{Y}(t)=\operatorname {E} [Y_{t}]} , then the cross-covariance is given by K X Y ⁡ ( t 1 , t 2 ) = cov ⁡ ( X t 1 , Y t 2 ) = E ⁡ [ ( X t 1 − μ X ( t 1 ) ) ( Y t 2 − μ Y ( t 2 ) ) ] = E ⁡ [ X t 1 Y t 2 ] − μ X ( t 1 ) μ Y ( t 2 ) . {\displaystyle \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}).\,} Cross-covariance is related to the more commonly used cross-correlation of the processes in question. In the case of two random vectors X = ( X 1 , X 2 , … , X p ) T {\displaystyle \mathbf {X} =(X_{1},X_{2},\ldots ,X_{p})^{\rm {T}}} and Y = ( Y 1 , Y 2 , … , Y q ) T {\displaystyle \mathbf {Y} =(Y_{1},Y_{2},\ldots ,Y_{q})^{\rm {T}}} , the cross-covariance would be a p × q {\displaystyle p\times q} matrix K X Y {\displaystyle \operatorname {K} _{XY}} (often denoted cov ⁡ ( X , Y ) {\displaystyle \operatorname {cov} (X,Y)} ) with entries K X Y ⁡ ( j , k ) = cov ⁡ ( X j , Y k ) . {\displaystyle \operatorname {K} _{XY}(j,k)=\operatorname {cov} (X_{j},Y_{k}).\,} 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.