On the comparison of diversity of parts of a distribution

Rajaram, Rajeev; Ritchey, Nathan; Castellani, Brian

Abstract

The literature on diversity measures, regardless of the metric used (e.g.,Gini-Simpson index, Shannon entropy) has a notable gap: not much has been done to connect these measures back to the shape of the original distribution, or to use them to compare the diversity of parts of a given distribution and their relationship to the diversity of the whole distribution. As such, the precise quantification of the relationship between the probability of each type pi and the diversity Din non-uniform distributions, both among parts
of a distribution as well as the whole, remains unresolved. This is particularly true for Hill numbers, despite their usefulness as‘effective numbers’. This gap is problematic as most real-world systems(e.g., income distributions, economic complexity indices, rankings, ecological systems) have unequal distributions, varying frequencies, and comprise multiple diversity types with unknown frequencies that can change. To address this issue, we connect case-based entropy, an approach to diversity we developed, to the shape of a
probability distribution; allowing us to show that the original probability distribution g1, the case-based entropy curve g2 and the c{1,k} versus the c{1,k} *ln A {1,k} curve g3, which we call the slope of diversity, are one-to-one (or injective), i.e., a different probability distribution g1 gives a different curvefor g2 and g3. Hence, a different permutation of the original probability distribution g1(that leads to a different shape) will uniquely determine the graphs g2 and g3. By proving the injective nature of our approach, we will have
established a unique way to measure the degree of uniformity of parts as measured byDP/cP for a given part P of the original probability distribution, and also have shown a unique way to compute theDP/cP for various shapes of the original distribution and (in terms of comparison)for different curves.

Citation

Rajaram, R., Ritchey, N., & Castellani, B. (2023). On the comparison of diversity of parts of a distribution. Journal of Physics Communications, 7(7), Article 075006. https://doi.org/10.1088/2399-6528/ace952

• Journal Article Type: Article
• Acceptance Date: Jul 20, 2023
• Online Publication Date: Aug 1, 2023
• Publication Date: 2023
• Deposit Date: Dec 11, 2023
• Publicly Available Date: Dec 11, 2023
• Journal: Journal of Physics Communications
• Electronic ISSN: 2399-6528
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 7
• Issue: 7
• Article Number: 075006
• DOI: https://doi.org/10.1088/2399-6528/ace952
• Keywords: General Physics and Astronomy
• Public URL: https://durham-repository.worktribe.com/output/2022781