Cross-covariance

Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
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In probability and statistics, given two stochastic processes ${\displaystyle \left\{X_{t}\right\}}$ and ${\displaystyle \left\{Y_{t}\right\}}$, 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 ${\displaystyle \operatorname {E} }$ for the expectation operator, if the processes have the mean functions ${\displaystyle \mu _{X}(t)=\operatorname {\operatorname {E} } [X_{t}]}$ and ${\displaystyle \mu _{Y}(t)=\operatorname {E} [Y_{t}]}$, then the cross-covariance is given by

${\displaystyle \operatorname {K} _{XY}(t_{1},t_{2})=\operatorname {cov} (X_{t_{1}},Y_{t_{2}})=\operatorname {E} [(X_{t_{1}}-\mu _{X}(t_{1}))(Y_{t_{2}}-\mu _{Y}(t_{2}))]=\operatorname {E} [X_{t_{1}}Y_{t_{2}}]-\mu _{X}(t_{1})\mu _{Y}(t_{2}).\,}$

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

In the case of two random vectors ${\displaystyle \mathbf {X} =(X_{1},X_{2},\ldots ,X_{p})^{\rm {T}}}$ and ${\displaystyle \mathbf {Y} =(Y_{1},Y_{2},\ldots ,Y_{q})^{\rm {T}}}$, the cross-covariance would be a ${\displaystyle p\times q}$ matrix ${\displaystyle \operatorname {K} _{XY}}$ (often denoted ${\displaystyle \operatorname {cov} (X,Y)}$) with entries ${\displaystyle \operatorname {K} _{XY}(j,k)=\operatorname {cov} (X_{j},Y_{k}).\,}$ Thus the term cross-covariance is used in order to distinguish this concept from the covariance of a random vector ${\displaystyle \mathbf {X} }$, which is understood to be the matrix of covariances between the scalar components of ${\displaystyle \mathbf {X} }$ 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.