While recent work has established divergence as a key framework for understanding evenness, there is currently no research exploring how the families of measures within the divergence-based framework relate to each other. This paper uses geometry to show that, holding order and richness constant, the families of divergence-based evenness measures nest. This property allows them to be ranked based on their reactivity to changes in relatively even assemblages or changes in relatively uneven ones. We establish this ranking and explore how the distance-based measures relate to it for both order q = 2 and q = 1. We also derive a new family of distance-based measures that captures the angular distance between the vector of relative abundances and a perfectly even vector and is highly reactive to changes in even assemblages. Finally, we show that if we only require evenness to be a divergence, then any smooth, monotonically increasing function of diversity can be made into an evenness measure. A deeper understanding of how to measure evenness will require empirical or theoretical research that uncovers which kind of divergence best reflects the underlying concept.

虽然最近的工作已将散度确定为理解均匀度的关键框架，但目前还没有研究探索基于散度的框架内的测量系列如何相互关联。本文使用几何证明，在保持秩序和丰富度不变的情况下，基于散度的均匀度测度族是嵌套的。此属性允许根据它们对相对均匀组合中的变化或相对不均匀组合中的变化的反应性对它们进行排名。我们建立了这个排名并探索了基于距离的度量如何与 q = 2 和 q = 1 的顺序相关。我们还推导出了一个新的基于距离的度量系列，它捕获了相对丰度向量和 a 之间的角距离。完全均匀的向量，并且对均匀组合的变化具有高度反应性。最后，我们表明，如果我们只要求均匀度是一个散度，那么任何平滑的、单调递增的多样性函数都可以成为一个均匀度测度。更深入地了解如何测量均匀度将需要实证或理论研究，以揭示哪种差异最能反映基本概念。