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 of random vectors
Cross-covariance of stochastic processes
The definition of cross-covariance of random vector may be generalized to stochastic processes as follows:
Let and denote stochastic processes. Then the cross-covariance function of the processes is defined by::p.172
where and .
If the processes are complex stochastic processes, the second factor needs to be complex conjugated.
Definition for jointly WSS processes
If and are a jointly wide-sense stationary, then the following are true:
- for all ,
- for all
- 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.:p.142 Formally:
Cross-covariance of deterministic signals
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). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a sub-sampling of one of the signals). For a large number of samples, the average converges to the true covariance.
Cross-covariance may also refer to a "deterministic" cross-covariance between two signals. This consists of summing over all time indices.
For example, for discrete-time signals and the cross-covariance is defined as
where the line indicates that the complex conjugate is taken when the signals are complex-valued.
For continuous functions and the (deterministic) cross-covariance is defined as
The (deterministic) cross-covariance of two continuous signals is related to the convolution by
and the (deterministic) cross-covariance of two discrete-time signals is related to the discrete convolution by
- ^ a b Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3