Abstract-In computing the similarity of intervals, current similarity measures such as the commonly used Jaccard and Dice measures are at times not sensitive to changes in the width of intervals, producing equal similarities for substantially different pairs of intervals. To address this, we propose a new similarity measure that uses a bi-directional approach to determine interval similarity. For each direction, the overlapping ratio of the given interval in a pair with the other interval is used as a measure of uni-directional similarity. We show that the proposed measure satisfies all common properties of a similarity measure, while also being invariant in respect to multiplication of the interval endpoints and exhibiting linear growth in respect to linearly increasing overlap. Further, we compare the behavior of the proposed measure with the highly popular Jaccard and Dice similarity measures, highlighting that the proposed approach is more sensitive to changes in interval widths. Finally, we show that the proposed similarity is bounded by the Jaccard and the Dice similarity, thus providing a reliable alternative.

With a growing number of areas leveraging intervalvalued data-including in the context of modelling 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 settheoretic 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 measurestheir 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. 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 intuitionessential 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. Index Terms-similarity measure, distance measure, subsethood, interval-valued data 1 Beyond both Jaccard and Dice measures, other set-theoretic SMs such as Szymkiewicz-Simpson coefficient [10], Otsuka coefficient [11], Sokal-Sneath Coefficient [12], Simple-Matching coefficient [13] are also subject to aliasing for intervals.

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Similarity measures are useful for reasoning about fuzzy sets. Hence, many classical set-theoretic similarity measures have been extended for comparing fuzzy sets. In previous work, a set-theoretic similarity measure considering the bidirectional subsethood for intervals was introduced. The measure addressed specific concerns of many common similarity measures, and it was shown to be bounded above and below by Jaccard and Dice measures respectively. Herein, we extend our prior measure from similarity on intervals to fuzzy sets. Specifically, we propose a vertical-slice extension where two fuzzy sets are compared based on their membership values. We show that the proposed extension maintains all common properties (i.e., reflexivity, symmetry, transitivity, and overlapping) of the original fuzzy similarity measure. We demonstrate and contrast its behaviour along with common fuzzy set-theoretic measures using different types of fuzzy sets (i.e., normal, non-normal, convex, and non-convex) in respect to different discretization levels.

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