A note on distance variance for categorical variables

TL;DR

This paper explores extending distance variance to categorical variables, revealing its connections to entropy but also its limitations due to failure in satisfying Schur-concavity for variables with multiple categories.

Contribution

It provides geometric and algebraic characterizations of distance variance for categorical variables and identifies its limitations in handling multi-category cases.

Findings

Distance variance relates to entropy measures.

Fails Schur-concavity for variables with >2 categories.

Limitations restrict its universal applicability.

Abstract

This study investigates the extension of distance variance, a validated spread metric for continuous and binary variables [Edelmann et al., 2020, Ann. Stat., 48(6)], to quantify the spread of general categorical variables. We provide both geometric and algebraic characterizations of distance variance, revealing its connections to some commonly used entropy measures, and the variance-covariance matrix of the one-hot encoded representation. However, we demonstrate that distance variance fails to satisfy the Schur-concavity axiom for categorical variables with more than two categories, leading to counterintuitive results. This limitation hinders its applicability as a universal measure of spread.