On Nested Diversities and Novel Correlation-Based Entropies

The notion of diversity spans across disciplines, arising in a wide range of natural and social sciences. As is well-acknowledged, essential defining characteristics of diversity include the number of categories involved, evenness of the element distribution among categories, and dissimilarity between categories. In addition, there are two intrinsic factors that also influence the quantification of diversity. These are how much weight to place on the category prevalence, and how much weight to place on the between-category dissimilarities. In Part I of this paper, we provide a unified framework for measuring, comparing and partitioning diversity that incorporates all of these aspects. A new principle called the Nesting Principle is established, stating that diversity remains invariant under arbitrary nesting or partitioning of the constituent categories or individual elements. An important consequence of this principle is that the weight on the category prevalence must be set such that the diversity function depends quadratically on the relative abundances of categories, while the other weight on the between-category dissimilarity can be adjusted independently and continuously. The resulting diversity index has a unique interpretation in terms of a certain correlation sum defined within the assemblage whose diversity is to be measured. The formula for the effective dissimilarity between two arbitrary assemblages is also provided, which takes the form of a cross-correlation sum, and is compatible with the Nesting Principle. In ecology, this provides a possible solution to the long-standing problem of how to partition diversity into within and between components in a consistent manner. Classic entropy measures of Rényi and Tsallis, including Boltzmann-Gibbs and Shannon entropy, are also generalised to incorporate the dependence on correlations among the constituents. In Part II, we discuss applications of the correlation-based, nesting-invariant entropies to specific research contexts. Specifically, the extended Rényi entropy is used to analyse dynamical and fractal properties of a chaotic attractor system. Further, implications of the extended Tsallis entropy on non-extensive statistical physics are explored. ∗ k.okamura_at_alumni.lse.ac.uk ar X iv :1 80 4. 02 45 4v 1 [ qbi o. Q M ] 2 9 M ar 2 01 8