Cross covariance

In statistics, the term "cross-covariance" is sometimes used to refer to the covariance cov("X", "Y") between two random vectors "X" and "Y", in order to distinguish that concept from the "covariance" of a random vector "X", which is understood to be the matrix of covariances between the scalar components of "X".

In signal processing, the cross-covariance (or sometimes "cross-correlation") 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.

For discrete functions "f"_"i" and "g"_"i" the cross-covariance is defined as

:$(fstar\; g)\_i\; stackrel\{mathrm\{def\{=\}\; sum\_j\; f^*\_j,g\_\{i+j\}$

where the sum is over the appropriate values of the integer "j" and an asterisk indicates the complex conjugate. For continuous functions "f" (x) and "g" _i the cross-covariance is defined as

:$(fstar\; g)(x)\; stackrel\{mathrm\{def\{=\}\; int\; f^*(t)\; g(x+t),dt$

where the integral is over the appropriate values of "t".

The cross-covariance is similar in nature to the convolution of two functions.

Properties

The cross-covariance is related to the convolution by:

:$f(t)star\; g(t)\; =\; f^*(-t)*g(t)$

so that

:$(fstar\; g)\; =\; f*g$

if either f or g is an even function. Also:

:$(fstar\; g)star(fstar\; g)=(fstar\; f)star\; (gstar\; g)$

ee also

* Convolution
* Correlation
* Autocovariance

External links

* [http://mathworld.wolfram.com/Cross-Correlation.html Cross Correlation from Mathworld]
* http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:physics/0405041
* http://scribblethink.org/Work/nvisionInterface/nip.html
* http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
* http://archive.nlm.nih.gov/pubs/hauser/Tompaper/tompaper.php
* http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf