Further Generalizations of the Jaccard Index
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Quantifying the similarity between two sets constitutes a particularly interesting and useful operation in several theoretical and applied problems involving set theory. Aimed at quantifying the similarity between two sets, the Jaccard index has been extensively used in the most diverse types of problems, also motivating respective generalizations. The present work addressew further generalizations of this index, including its modification into a coincidence index capable of accounting also for the level of interiority of the sets, an extension for sets in continuous vector spaces, the consideration of weights associated to the involved set elements, the generalization to multiset addition, densities and generic scalar fields, as well as a means to quantify the joint interdependence between random variables. The also interesting possibility to take into account more than two sets was also addressed, including the description of an index capable of quantifying the level of chaining between three sets. Several of the described and suggested generalizations 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.
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