Further Generalizations of the Jaccard Index

TL;DR

This paper introduces new generalizations of the Jaccard index, including modifications for continuous spaces, multiset addition, densities, scalar fields, and joint interdependence, enhancing its applicability in data analysis and pattern recognition.

Contribution

It presents novel extensions of the Jaccard index for various mathematical structures and data types, broadening its use in theoretical and applied contexts.

Findings

Introduced a coincidence index accounting for interiority levels.

Extended the index to continuous vector spaces and scalar fields.

Illustrated the generalizations with numeric examples.

Abstract

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.