Using Geometry to Rank Evenness Measures: Towards a Deeper Understanding of Divergence

TL;DR

This paper uses geometric methods to analyze divergence-based evenness measures, revealing their nested relationships, and introduces a new angular distance measure that is highly sensitive to changes in evenness.

Contribution

It demonstrates the nesting property of divergence-based evenness measures and introduces a novel angular distance measure for assessing evenness.

Findings

Divergence-based measures nest when holding order and richness constant.

A new angular distance measure effectively captures changes in evenness.

Any smooth, increasing function of diversity can serve as an evenness measure under divergence.

Abstract

While recent work has established divergence as a key framework for understanding evenness, there is currently no research exploring how the families of measures within the divergence-based framework relate to each other. This paper uses geometry to show that, holding order and richness constant, the families of divergence-based evenness measures nest. This property allows them to be ranked based on their reactivity to changes in relatively even assemblages or changes in relatively uneven ones. We establish this ranking and explore how the distance-based measures relate to it for both order q=2 and q=1. We also derive a new family of distance-based measures that captures the angular distance between the vector of relative abundances and a perfectly even vector and is highly reactive to changes in even assemblages. Finally, we show that if we only require evenness to be a divergence, then any smooth, monotonically increasing function of diversity can be made into an evenness measure. A deeper understanding of how to measure evenness will require empirical or theoretical research that uncovers which kind of divergence best reflects the underlying concept.