Decomposition of Whittaker’s gamma diversity: a novel way combining entropies and divergences

Abstract

Accurate, standardized, and comparable methods for estimating biodiversity are crucial in ecology to properly assess and monitor the health of communities. Special cases of generalized entropy are commonly used to estimate alpha diversity. The related concept of generalized divergence can be used to estimate the beta diversity. Using cross entropy notion, we propose a modular decomposition of gamma diversity by using entropy and divergence functions. We prove that if alpha is Shannon entropy and beta is Kullback-Liebler divergence, the classical Whittaker’s gamma diversity is mathematically decomposed via our proposed local gamma index. To show the ecological application of this index and its generalization we compute the local gamma of several orders using a real large biological dataset. The index is discussed in detail for two limit cases, one where the contribution of rare species is the highest and one where richness and evenness are balanced. The index defines a gradient from communities that are dominated by a few common species toward samples shared among several uncommon ones. Our findings support divergence-based measures as practical estimators of beta diversity. Also, the framework here proposed, based on entropy, divergences and cross-entropies, allows us to compute the classic gamma diversity while providing components that are independent, comparable, self-reliant and pointwise distributed.