Distance correlation: Difference between revisions

Revision as of 02:50, 17 February 2015

Revision as of 07:39, 17 February 2015

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The classical measure of dependence, the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]],<ref>Pearson (1895)</ref> is mainly sensitive to a linear relationship between two variables. Distance correlation was introduced in 2005 by [[Gabor J Szekely]] in several lectures to address this deficiency of Pearson’s [[correlation]], namely that it can easily be zero for dependent variables. Correlation = 0 (uncorrelatedness) does not imply independence while distance correlation = 0 does imply independence. The first results on distance correlation were published in 2007 and 2009.<ref name=SR2007>Székely, Rizzo and Bakirov (2007)</ref><ref name=SR2009>Székely & Rizzo (2009)</ref> It was proved that distance covariance is the same as the Brownian covariance.<ref name=SR2009/> These measures are examples of [[energy distance]]s.

==Definitions==

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An unbiased estimator of <math>\operatorname{dCov}^2(X,Y)</math> is given in .<ref name=SR2014>Székely & Rizzo (2014).</ref>.

===Distance variance===

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==Alternative definition of distance covariance==

The original [[Distance_correlationDistance correlation#Distance_covariance_1|Distance covariance 1|distance covariance]] has been defined as the square root of <math>\operatorname{dCov}^2(X,Y)</math>, rather than the squared coefficient itself. <math>\operatorname{dCov}(X,Y)</math> has the property that it is the [[energy distance]] between the joint distribution of <math>\operatorname X, Y </math> and the product of its marginals. Under this definition, however, the distance variance, rather than the distance standard deviation, is measured in the same units as the scalar random variable <math>\operatorname X </math>.

Alternately, one could define '''''distance covariance''''' to be the square of the energy distance:

<math> \operatorname{dCov}^2(X,Y).</math>

In this case, there exists an unbiased estimator for the population distance covariance.<ref name=SR2014>Székely & Rizzo (2014)</ref>. An unbiased estimator does not exist for the coefficient <math>\operatorname{dCov}(X,Y).</math>

'''''Generalized distance covariance''''' could alternately be defined by the square, <math>\operatorname{dCov}^2(X, Y; \alpha) </math>, and similarly in this case an unbiased estimator of this generalized coefficient exists.