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Advancing Shannon Entropy for Measuring Diversity in Systems 时间:2017-06-27 13:52 来源:未知作者:admin 点击: 次 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.
1. Diversity in Systems
Statistical distributions play an important role in any branch of science that studies systems comprised of many similar or identical particles, objects, or actors, whether material or immaterial, human or nonhuman. One of the key features that determines the characteristics and range of potential behaviors of such systems is the degree and distribution of diversity, that is, the extent to which the components of the system occupy states with similar or different features.
As Page outlined in a series of inquiries [1, 2], including The Difference and Diversity and Complexity, diversity within systems is an important concern for science, be it making sense of economic inequality, expanding the trade portfolio of countries, measuring the collapse of species diversity in various ecosystems, or determining the optimal utility/robustness of a network. However, an important major challenge in the literature on diversity and complexity, which Page also points out [1, 2], remains: the issue of measurement. Although statistical distributions that directly reflect the spread of key parameters (such as mass, age, wealth, or energy) provide descriptions of this diversity, it can be difficult to compare the diversity of different distributions or even the same distribution under different conditions, mostly because of differences in scales and parameters. Also, many of the measures currently available compress diversity into a single score or are not intuitive [1–4].
At the outset, motivated by examples of measuring diversity in ecology and evolutionary biology from [3, 4], we sought to address these challenges. We begin with some definitions and a review of our previous research.
First, in terms of definitions, we follow the ecological literature, defining diversity as the interplay of “richness” and “evenness” in a probability distribution. Richness refers to the number of different diversity types in a system. Examples include (a) the different levels of household income in a city, (b) the number of different species in an ecosystem, (c) the diversity of a country’s exports, (d) the distribution of different nodes in a complex network, (e) the various health trends for a particular disease across time/space, or (f) the cultural or ethnic diversity of an organization or company. In all such instances, the greater the number of diversity types (be these types discrete or continuous), the greater the degree of richness in a system. In the case of the current study, for example, richness was defined as the number of different energy states.
In turn, evenness refers to the uniformity or “equiprobability” of occurrence of such states. In terms of the above examples, evenness would be defined as (a) a city where household income was evenly distributed, (b) an ecosystem where the diversity of its species was equal in number, (c) a country with an even distribution of exports, (d) a complex network where all nodes had the same probability of occurrence, (e) a disease where all possible health trends were equiprobable, or (f) a company or organization where people of different cultural or ethnic backgrounds were evenly distributed. In the case of the current study, for example, evenness was defined as the uniformity or “equiprobability” of the occurrence of all possible energy states.