Distance correlation: Difference between revisions

Revision as of 14:37, 16 November 2017

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Distance correlation can be used to perform a [[Statistical hypothesis testing|statistical test]] of dependence with a [[permutation test]]. One first computes the distance correlation (involving the re-centering of Euclidean distance matrices) between two random vectors, and then compares this value to the distance correlations of many shuffles of the data.

Distance correlation can be put into an indirect relationship to the ordinary moments by an [[#Alternative formulation: Brownian covariance|alternative formulation]] using ideas related to [[Brownian motion]]. This has led to the use of names such as '''Brownian covariance''' and '''Brownian distance covariance'''. Other correlational metrics, including kernel-based correlational metrics (such as the Hilbert-Schmidt Independence Criterion or HSIC) can also detect linear and nonlinear interactions. Both distance correlation and kernel-based metrics can be used in methods such as [[canonical correlation analysis]] and [[independent component analysis]] to yield stronger [[statistical power]].

[[Image:Distance Correlation Examples.svg|thumb|400px|right|Several sets of (''x'', ''y'') points, with the distance correlation coefficient of ''x'' and ''y'' for each set. Compare to the graph on [[correlation]]]]

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\operatorname{cov}_{\mathrm{id}}(X,Y) = \left\vert\operatorname{cov}(X,Y)\right\vert.

</math>

==Related metrics==

Distance correlation can be put into an indirect relationship to the ordinary moments by an [[#Alternative formulation: Brownian covariance|alternative formulation]] using ideas related to [[Brownian motion]]. This has led to the use of names such as '''Brownian covariance''' and '''Brownian distance covariance'''. Other correlational metrics, including kernel-based correlational metrics (such as the Hilbert-Schmidt Independence Criterion or HSIC) can also detect linear and nonlinear interactions. Both distance correlation and kernel-based metrics can be used in methods such as [[canonical correlation analysis]] and [[independent component analysis]] to yield stronger [[statistical power]].

==See also==

* [[RV coefficient]]

* For a related third-order statistic, see [[Skewness#Distance skewness|Distance skewness]].

*

==Notes==