Measuring Similarity Between Discontinuous Intervals - Challenges and Solutions

Discontinuous intervals (DIs) arise in a wide range of contexts, from real world data capture of human opinion to α-cuts of non-convex fuzzy sets. Commonly, for assessing the similarity of DIs, the latter are converted into their continuous form, followed by the application of a continuous interval (CI) compatible similarity measure. While this conversion is efficient, it involves the loss of discontinuity information and thus limits the accuracy of similarity results. Further, most similarity measures including the most popular ones, such as Jaccard and Dice, suffer from aliasing, that is, they are liable to return the same similarity for very different pairs of CIs. To address both of these challenges, this paper proposes a generalized approach for calculating the similarity of DIs which leverages the recently introduced bidirectional subsethood based similarity measure (which avoids aliasing) while accounting for all pairs of the continuous subintervals within the DIs to be compared. We provide detail of the proposed approach and demonstrate its behaviour when applying bidirectional subsethood, Jaccard and Dice as similarity measures, using different pairs of synthetic DIs. The experimental results show that the similarity outputs of the new generalized approach follow intuition for all three similarity measures; however, it is only the proposed integration with the bidirectional subsethood similarity measure which also avoids aliasing for DIs.