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Concept
Cross-covariance
Résumé
In probability and statistics, given two stochastic processes and , 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 for the expectation operator, if the processes have the mean functions and , 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 and , the cross-covariance would be a matrix (often denoted ) with entries Thus the term cross-covariance is used in order to distinguish this concept from the covariance of a random vector , which is understood to be the matrix of covariances between the scalar components of 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.
Cross-covariance matrix
The definition of cross-covariance of random vectors may be generalized to stochastic processes as follows:
Let and denote stochastic processes. Then the cross-covariance function of the processes is defined by:
where and .
If the processes are complex-valued stochastic processes, the second factor needs to be complex conjugated:
If and are a jointly wide-sense stationary, then the following are true:
for all ,
for all
and
for all
By setting (the time lag, or the amount of time by which the signal has been shifted), we may define
The cross-covariance function of two jointly WSS processes is therefore given by:
which is equivalent to
Two stochastic processes and are called uncorrelated if their covariance is zero for all times. Formally:
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).
Source officielle
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