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Advancing Shannon Entropy for Measuring Diversity in Systems

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  • R. Rajaram
  • B. Castellani
  • A. N. Wilson

Abstract

From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy has been used to rethink diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon’s approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a renormalization of probability distributions based on the notion of case-based entropy as a function of the cumulative probability . Given a probability density , measures the diversity of the distribution up to a cumulative probability of , by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of up to cumulative probability . We illustrate the utility of our approach by renormalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for energy of subatomic particles. The comparison shows that is a vast improvement over as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions.

Suggested Citation

  • R. Rajaram & B. Castellani & A. N. Wilson, 2017. "Advancing Shannon Entropy for Measuring Diversity in Systems," Complexity, Hindawi, vol. 2017, pages 1-10, May.
  • Handle: RePEc:hin:complx:8715605
    DOI: 10.1155/2017/8715605
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    References listed on IDEAS

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    1. Rajaram, R. & Castellani, B., 2016. "An entropy based measure for comparing distributions of complexity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 35-43.
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    Cited by:

    1. Md Amiruzzaman & Ye Zhao & Stefanie Amiruzzaman & Aryn C. Karpinski & Tsung Heng Wu, 2023. "An AI-based framework for studying visual diversity of urban neighborhoods and its relationship with socio-demographic variables," Journal of Computational Social Science, Springer, vol. 6(1), pages 315-337, April.
    2. Wenjie Hu & Hua Zhao & Tao Dong, 2018. "Dynamic Analysis for a Kaldor–Kalecki Model of Business Cycle with Time Delay and Diffusion Effect," Complexity, Hindawi, vol. 2018, pages 1-11, January.
    3. Sina Salimian & Seyed Meysam Mousavi & Jurgita Antucheviciene, 2022. "An Interval-Valued Intuitionistic Fuzzy Model Based on Extended VIKOR and MARCOS for Sustainable Supplier Selection in Organ Transplantation Networks for Healthcare Devices," Sustainability, MDPI, vol. 14(7), pages 1-21, March.

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