Ectropy of diversity measures for populations in Euclidean space

Measures to evaluate the diversity of a set of points (population) in Euclidean space play an important role in a variety of areas of science and engineering. Well-known measures are often used without a clear insight into their quality and many of them do not appropriately penalize populations with a few distant groups of collocated or closely located points. To the best of our knowledge, there is a lack of rigorous criteria to compare diversity measures and help select an appropriate one. In this work we define a mathematical notion of ectropy for classifying diversity measures in terms of the extent to which they tend to penalize point collocation, we investigate the advantages and disadvantages of several known measures and we propose some novel ones that exhibit a good ectropic behavior. In particular, we introduce a quasi-entropy measure based on a geometric covering problem, three measures based on discrepancy from uniform distribution and one based on Euclidean minimum spanning trees. All considered measures are tested and compared on a large set of random and structured populations. Special attention is also devoted to the complexity of computing the measures. Most of the novel measures compare favorably with the classical ones in terms of ectropy. The measure based on Euclidean minimum spanning trees turns out to be the most promising one in terms of the tradeoff between the ectropic behavior and the computational complexity.