Distance Correlation
Gábor J. Székely, Maria L. Rizzo · 2022
In the analysis of real data, statisticians often work with random vectors that may have non-linear dependence. In this chapter, we focus on dependence coefficients distance covariance (dCov) and distance correlation (dCor) introduced in Székely, Rizzo, and Bakirov [2007] that measure all types of dependence between random vectors X and Y in arbitrary dimension. These energy dependence coefficients characterize independence between random vectors: https://www.w3.org/1998/Math/MathML"> dCov ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429157158/120f0130-0a52-40d7-ac32-31c2dd83f69b/content/math12_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> (and https://www.w3.org/1998/Math/MathML"> dCor ( X , Y ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429157158/120f0130-0a52-40d7-ac32-31c2dd83f69b/content/math12_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> ) is non-negative and equal to zero if and only if the random vectors X and Y are independent. Distance correlation is a very effective tool to detect novel associations in large data sets (see Simon and Tibshirani [2011] and Wahba [2014] ).