Quantifying the similarity between two mathematical structures or datasets
constitutes a particularly interesting and useful operation in several
theoretical and applied problems. Aimed at this specific objective, the Jaccard
index has been extensively used in the most diverse types of problems, also
motivating some respective generalizations. The present work addresses further
generalizations of this index, including its modification into a coincidence
index capable of accounting also for the level of relative interiority between
the two compared entities, as well as respective extensions for sets in
continuous vector spaces, the generalization to multiset addition, densities
and generic scalar fields, as well as a means to quantify the joint
interdependence between two random variables. The also interesting possibility
to take into account more than two sets has also been addressed, including the
description of an index capable of quantifying the level of chaining between
three structures. Several of the described and suggested eneralizations have
been illustrated with respect to numeric case examples. It is also posited that
these indices can play an important role while analyzing and integrating
datasets in modeling approaches and pattern recognition activities, including
as a measurement of clusters similarity or separation and as a resource for
representing and analyzing complex networks.

对 Jaccard 指数及其相关的一些扩展进行理论研究，包括一种新的巧合指数，该指数在比较两个实体集时可以考虑相对内含度的水平，适用于连续向量空间、多重集、密度和通用标量场的扩展，以及量化两个随机变量之间的联合相互依赖性的方法。