A Similarity Measure Based on Bidirectional Subsethood for Intervals

Abstract

With a growing number of areas leveraging interval-valued data - including in the context of modeling human uncertainty (e.g., in cybersecurity), 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 Sorensen-Dice measures, are commonly used, whereas axiomatically, a wide breadth of possible measures have been theoretically explored. This article 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 traditional and new axioms. The proposed measure is designed to be sensitive to the variation in the size of intervals, thus avoiding aliasing. This article provides a detailed theoretical exploration of the new proposed measure, and systematically demonstrates its behavior 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 Sorensen-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