Distance standard deviation

In statistics and in probability theory, distance correlation is a measure of statistical dependence between two random variables or two random vectors of arbitrary, not necessarily equal dimension. It is zero if and only if the random variables are statistically independent, contrary to Pearson's correlation, which can be zero for dependent random variables.

The distance correlation is derived from a number of other quantities that are used in its specification, specifically: distance variance, distance standard deviation and distance covariance. These quantities take the same roles as the ordinary moments with corresponding names in the specification of the Pearson product-moment correlation coefficient.

These distance-based measures can be put into an indirect relationship to the ordinary moments by an alternative formulation (described below) using ideas related to Brownian motion, and this has led to the use of names such as Brownian covariance and Brownian distance covariance.

The classical measure of dependence, the Pearson correlation coefficient, 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. It was proved that distance covariance is the same as the Brownian covariance. These measures are examples of energy distances.

...
Wikipedia