Abstract
In order to study the spread of an epidemic over a region as a function of time, we introduce an entropy ratio U describing the uniformity of infections over various states and their districts, and an entropy concentration coefficient The latter is a multiplicative version of the Kullback-Leibler distance, with values between 0 and 1. For product measures and self-similar phenomena, it does not depend on the measurement level. Hence, C is an alternative to Gini’s concentration coefficient for measures with variation on different levels. Simple examples concern population density and gross domestic product. Application to time series patterns is indicated with a Markov chain. For the Covid-19 pandemic, entropy ratios indicate a homogeneous distribution of infections and the potential of local action when compared to measures for a whole region.
Highlights
In any statistical data analysis, a first step is to determine two parameters that characterize center and variation of the data
Covid-19 was the motivation for our study, it turns out that entropy ratio and entropy concentration coefficient are appropriate parameters for many other phenomena
We study the concentration of the gross domestic product (GDP) among countries, and among collections of countries defined by the UN as sustainable development goal regions
Summary
In any statistical data analysis, a first step is to determine two parameters that characterize center (mean, median) and variation of the data. Shannon entropy [3] and the relative entropy introduced by Kullback and Leibler [4] have been used as measures of uniform distribution and concentration Their values grow with the size of the data, which makes interpretation difficult. Covid-19 was the motivation for our study, it turns out that entropy ratio and entropy concentration coefficient are appropriate parameters for many other phenomena. The paper is completed with conclusions and Appendix A on the Gini coefficient
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