Distance correlation: Difference between revisions

Revision as of 00:54, 17 February 2015

Revision as of 02:50, 17 February 2015

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==Alternative definition of distance covariance==

The original [[Distance_correlation#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:

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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/\alpha}(X, Y; \alpha) </math>, and similarly in this case an unbiased estimator of this generalized coefficient exists.

Under these alternate definitions, the distance correlation is also defined as the square <math>\operatorname{dCor}^2(X,Y)</math>, rather than the square root.