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A Similarity Measure Based on Bidirectional Subsethood for Intervals

Kabir, Shaily; Wagner, Christian; Havens, Timothy C.; Anderson, Derek T.


Shaily Kabir

Timothy C. Havens

Derek T. Anderson


With a growing number of areas leveraging interval-valued data—including in the context of modelling human uncertainty (e.g., in Cyber Security), the capacity to accurately and systematically compare intervals for reasoning and computation is increasingly important. In practice, well established set-theoretic similarity measures such as the Jaccard and Sørensen-Dice measures are commonly used, while axiomatically a wide breadth of possible measures have been theoretically explored. This paper identifies, articulates, and addresses an inherent and so far not discussed limitation of popular measures—their tendency to be subject to aliasing—where they return the same similarity value for very different sets of intervals. The latter risks counter-intuitive results and poor automated reasoning in real-world applications dependent on systematically comparing interval-valued system variables or states. Given this, we introduce new axioms establishing desirable properties for robust similarity measures, followed by putting forward a novel set-theoretic similarity measure based on the concept of bidirectional subsethood which satisfies both the traditional and new axioms. The proposed measure is designed to be sensitive to the variation in the size of intervals, thus avoiding aliasing. The paper provides a detailed theoretical exploration of the new proposed measure, and systematically demonstrates its behaviour using an extensive set of synthetic and real-world data. Specifically, the measure is shown to return robust outputs that follow intuition—essential for real world applications. For example, we show that it is bounded above and below by the Jaccard and Sørensen-Dice similarity measures (when the minimum t-norm is used). Finally, we show that a dissimilarity or distance measure, which satisfies the properties of a metric, can easily be derived from the proposed similarity measure.


Kabir, S., Wagner, C., Havens, T. C., & Anderson, D. T. (2020). A Similarity Measure Based on Bidirectional Subsethood for Intervals. IEEE Transactions on Fuzzy Systems, 28(11), 2890-2904.

Journal Article Type Article
Acceptance Date Sep 12, 2019
Online Publication Date Mar 2, 2020
Publication Date 2020-11
Deposit Date Oct 8, 2019
Publicly Available Date Mar 2, 2020
Journal IEEE Transactions on Fuzzy Systems
Print ISSN 1063-6706
Electronic ISSN 1941-0034
Publisher Institute of Electrical and Electronics Engineers
Peer Reviewed Peer Reviewed
Volume 28
Issue 11
Pages 2890-2904
Keywords Control and Systems Engineering; Computational Theory and Mathematics; Applied Mathematics; Artificial Intelligence
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