Novel measures of symbol dominance (dC1 and dC2), symbol diversity (DC1 = N (1 − dC1) and DC2 = N (1 − dC2)), and information entropy (HC1 = log2 DC1 and HC2 = log2 DC2) are derived from Lorenz-consistent statistics that I had previously proposed to quantify dominance and diversity in ecology. Here, dC1 refers to the average absolute difference between the relative abundances of dominant and subordinate symbols, with its value being equivalent to the maximum vertical distance from the Lorenz curve to the 45-degree line of equiprobability; dC2 refers to the average absolute difference between all pairs of relative symbol abundances, with its value being equivalent to twice the area between the Lorenz curve and the 45-degree line of equiprobability; N is the number of different symbols or maximum expected diversity. These Lorenz-consistent statistics are compared with statistics based on Shannon’s entropy and Rényi’s second-order entropy to show that the former have better mathematical behavior than the latter. The use of dC1, DC1, and HC1 is particularly recommended, as only changes in the allocation of relative abundance between dominant (pd > 1/N) and subordinate (ps < 1/N) symbols are of real relevance for probability distributions to achieve the reference distribution (pi = 1/N) or to deviate from it.

中文翻译：

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洛伦兹曲线：定义符号优势、符号多样性和信息熵的令人满意的度量的适当框架

符号优势（dC1 和 dC2）、符号多样性（DC1 = N (1 − dC1) 和 DC2 = N (1 − dC2)）和信息熵（HC1 = log2 DC1 和 HC2 = log2 DC2）的新度量来自我之前提出的洛伦兹一致性统计量，用于量化生态学中的优势和多样性。其中，dC1是指主从符号相对丰度的平均绝对差，其值相当于洛伦兹曲线到45度等概率线的最大垂直距离；dC2 是指所有相对符号丰度对之间的平均绝对差值，其值相当于洛伦兹曲线与 45 度等概率线之间面积的两倍；N 是不同符号的数量或最大预期分集。将这些洛伦兹一致性统计量与基于香农熵和雷尼二阶熵的统计量进行比较，表明前者比后者具有更好的数学行为。特别推荐使用 dC1、DC1 和 HC1，因为只有显性 (pd > 1/N) 和从属 (ps < 1/N) 符号之间的相对丰度分配的变化才与概率分布实现真正相关参考分布 (pi = 1/N) 或偏离它。