What is "cross-covariance"

In probability and statistics, given two stochastic processes X = (X) and Y = (Y), 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 for the expectation operator, if the processes have the mean functions μ = E[X] and ν = E[Y], then the cross-covariance is given by

C(t, s) = cov(X, Y) = E[(X − μ)(Y − ν)] = E[XY] − μν. 

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, X, ..., X) and Y = (Y, Y, ..., Y), the cross-covariance would be a square n by n matrix C with entries C(j, k) = cov(X, Y).  Thus the term cross-covariance is used in order to distinguish this concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar components of 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.