Distance correlation: Difference between revisions

Revision as of 19:32, 7 March 2016

Revision as of 20:18, 7 March 2016

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==Background==

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{{cite|author=G. J. Szekely, M. L. Rizzo, and N. K. Bakirov| (year=2007)| title= Measuring and Testing Independence by Correlation of Distances| journal= Annals of Statistics| volume=35| issue=6| pages=2769-2794| url=http://dx.doi.org/10.1214/009053607000000505}}.</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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</math>

Then for every <math>0<\alpha<2</math>, <math>X</math> and <math>Y</math> are independent if and only if <math>\operatorname{dCov}^2(X, Y; \alpha) = 0</math>. It is important to note that this characterization does not hold for exponent <math>\alpha=2</math>; in this case for bivariate <math>(X, Y)</math>, <math>\operatorname{dCor}(X, Y; \alpha=2)</math> is a deterministic function of the Pearson correlation.<ref name=SR2007>Székely & Rizzo (2007) Theorem 7, p. 2785.</ref> If <math>a_{k,\ell}</math> and <math>b_{k,\ell}</math> are <math>\alpha</math> powers of the corresponding distances, <math>0<\alpha\leq2</math>, then <math>\alpha</math> sample distance covariance can be defined as the nonnegative number for which

:<math>

\operatorname{dCov}^2_n(X, Y; \alpha):= \frac{1}{n^2}\sum_{k,\ell}A_{k,\ell}\,B_{k,\ell}.

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==References==

*Bickel, P.J. and Xu, Y. (2009) "Discussion of: Brownian distance covariance", ''Annals of Applied Statistics'', 3 (4), 1266&ndash;1269. {{doi|10.1214/09-AOAS312A}} [http://arxiv4projecteuclid.library.cornell.eduorg/PS_cachedownload/arxivpdfview_1/pdfeuclid.aoas/0912/0912.3295v2.1267453934 pdf Free access to article]

*Gini, C. (1912). Variabilità e Mutabilità. Bologna: Tipografia di Paolo Cuppini.

*Pearson, K. (1895). "Note on regression and inheritance in the case of two parents", ''[[Proceedings of the Royal Society]]'', 58, 240&ndash;242

*Pearson, K. (1920). "Notes on the history of correlation", ''[[Biometrika]]'', 13, 25&ndash;45.

* Székely, G. J. and Rizzo, M. L. (2009). "Brownian distance covariance", ''Annals of Applied Statistics'', 3/4, 1233&ndash;1303. {{doi| 10.1214/09-AOAS312}} [http://personalprojecteuclid.bgsu.eduorg/~mrizzodownload/energypdfview_1/AOAS312euclid.pdfaoas/1267453933 Reprintpdf]

*Kosorok, M. R. (2009) "Discussion of: Brownian Distance Covariance", ''Annals of Applied Statistics'', 3/4, 1270–1278. {{doi|10.1214/09-AOAS312B}} [http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.0822v10822v2.pdf Free access to articlepdf]

*Székely, G.J. and Rizzo, M.L. (2014) Partial distance correlation with methods for dissimilarities, The Annals of Statistics, 42/6, 2382-2412.[http://projecteuclid.org/euclid.aos/1413810731][http://arxiv.org/pdf/1310.2926.pdf pdf].

==External links==