How to measure multidimensional variation?

TL;DR

This paper introduces a new multidimensional coefficient of variation derived from the Gini index, connecting sparsity measures and extending univariate concepts to multivariate data.

Contribution

It proposes a novel multidimensional coefficient of variation based on the Gini index, maintaining key properties of the univariate version and relating to existing multivariate measures.

Findings

The proposed coefficient aligns with properties of the univariate coefficient of variation.

It establishes a connection with the Voinov-Nikulin coefficient.

The new measure is compared favorably with existing multivariate coefficients.

Abstract

The coefficient of variation, which measures the variability of a distribution from its mean, is not uniquely defined in the multidimensional case, and so is the multidimensional Gini index, which measures the inequality of a distribution in terms of the mean differences among its observations. In this paper, we connect these two notions of sparsity, and propose a multidimensional coefficient of variation based on a multidimensional Gini index. We demonstrate that the proposed coefficient possesses the properties of the univariate coefficient of variation. We also show its connection with the Voinov-Nikulin coefficient of variation, and compare it with the other multivariate coefficients available in the literature.