User:Mathstat/Partial distance correlation

In statistics and in probability theory, partial distance correlation is a measure of statistical dependence between two random variables or two random vectors controlling for or removing the effect of one or more random variables. This measure extends distance covariance and distance correlation in a similar sense that partial correlation extends correlation. The random variables/vectors of interest take values in arbitrary, not necessarily equal dimension Euclidean space.

Definitions
The sample partial distance covariance is defined in terms of orthogonal projections as follows.