Mathematical diversity of parts for a continuous distribution

Abstract

The current paper is part of a series exploring how to link diversity measures (e.g., Gini-Simpson index, Shannon entropy, Hill numbers) to a distribution’s original shape and to compare parts of a distribution, in terms of diversity, with the whole. This linkage is crucial to understanding the exact relationship between the density of an original probability distribution, denoted by p(x), and the diversity D in non-uniform distributions, both within parts of a distribution and the whole. Empirically, our results are an important advance since we can compare various parts of a distribution, noting that systems found in contemporary data often have unequal distributions that possess multiple diversity types and have unknown and changing frequencies at different scales (e.g. income, economic complexity ratings, rankings, etc.). To date, we have proven our results for discrete distributions. Our focus here is continuous distributions. In both instances, we do so by linking case-based entropy, a diversity approach we developed, to a probability distribution’s shape for continuous distributions. This allows us to demonstrate that the original probability distribution g 1, the case-based entropy curve g 2, and the slope of diversity g 3 (c (a,x) versus the c(a,x)*lnA(a,x) curve) are one-to-one (or injective). Put simply, a change in the probability distribution, g 1, leads to variations in the curves for g 2 and g 3. Consequently, any alteration in the permutation of the initial probability distribution, which results in a different form, will distinctly define the graphs g 2 and g3 . By demonstrating the injective property of our method for continuous distributions, we introduce a unique technique to gauge the level of uniformity as indicated by D/c. Furthermore, we present a distinct method to calculate D/c for different forms of the original continuous distribution, enabling comparison of various distributions and their components.