Part of a series on Statistics |
Correlation and covariance |
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![](//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/CorrelationIcon.svg/100px-CorrelationIcon.svg.png) |
For random vectors - Autocorrelation matrix
- Cross-correlation matrix
- Auto-covariance matrix
- Cross-covariance matrix
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For deterministic signals |
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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
![{\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}).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05f4077f8173bfb323a68313e4287d017cc0c281)
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 vectors may be generalized to stochastic processes as follows:
Definition
Let
and
denote stochastic processes. Then the cross-covariance function of the processes
is defined by:[1]: p.172
![{\displaystyle \operatorname {K} _{XY}(t_{1},t_{2}){\stackrel {\mathrm {def} }{=}}\ \operatorname {cov} (X_{t_{1}},Y_{t_{2}})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right)\left(Y(t_{2})-\mu _{Y}(t_{2})\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43a9557f62a2b87ad3e9ea3fe34fd2886a555452) | | (Eq.1) |
where
and
.
If the processes are complex-valued stochastic processes, the second factor needs to be complex conjugated:
![{\displaystyle \operatorname {K} _{XY}(t_{1},t_{2}){\stackrel {\mathrm {def} }{=}}\ \operatorname {cov} (X_{t_{1}},Y_{t_{2}})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right){\overline {\left(Y(t_{2})-\mu _{Y}(t_{2})\right)}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af132571e88ed273022677215e210816f4f9b93)
Definition for jointly WSS processes
If
and
are a jointly wide-sense stationary, then the following are true:
for all
,
for all ![{\displaystyle t_{1},t_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e76daf1a59dca26c96dbca2863a1c236b15b5a1)
and
for all ![{\displaystyle t_{1},t_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e76daf1a59dca26c96dbca2863a1c236b15b5a1)
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:
![{\displaystyle \operatorname {K} _{XY}(\tau )=\operatorname {cov} (X_{t},Y_{t-\tau })=\operatorname {E} [(X_{t}-\mu _{X})(Y_{t-\tau }-\mu _{Y})]=\operatorname {E} [X_{t}Y_{t-\tau }]-\mu _{X}\mu _{Y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d33105bfcc12b9bd9a3d240d78fad1df3cf524) | | (Eq.2) |
which is equivalent to
.
Uncorrelatedness
Two stochastic processes
and
are called uncorrelated if their covariance
is zero for all times.[1]: 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
![{\displaystyle (f\star g)[n]\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k\in \mathbb {Z} }{\overline {f[k]}}g[n+k]=\sum _{k\in \mathbb {Z} }{\overline {f[k-n]}}g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02f82f14c2e14226808b1b3517153c75716e658d)
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
.
Properties
The (deterministic) cross-covariance of two continuous signals is related to the convolution by
![{\displaystyle (f\star g)(t)=({\overline {f(-\tau )}}*g(\tau ))(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/299c13707750e3cc555edaa4d9f99f2e6b559832)
and the (deterministic) cross-covariance of two discrete-time signals is related to the discrete convolution by
.
See also
References
- ^ a b Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
External links
- Cross Correlation from Mathworld
- http://scribblethink.org/Work/nvisionInterface/nip.html
- http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
- http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf