A Bidirectional Subsethood Based Similarity Measure for Fuzzy Sets

Abstract

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.

Measuring Similarity Between Discontinuous Intervals - Challenges and Solutions

“…A new SM for the CIs was introduced in [16] which uses the reciprocal subsethoods [17] or overlapping ratios [16] of a pair of CIs for capturing their similarity. This measure for two CIs a and b [16] [17] is…”

“…However, a further problem, particularly, the aliasing issue with common SMs, such as Jaccard and Dice has recently been identified [16], where the same similarity is returned for very different sets of intervals. A recently introduced SM for CIs using their overlapping ratios [16], also called bidirectional subsethood [17] has been shown to avoid aliasing for CIs.…”

“…In this paper, we propose a generalized SM for DIs which combines the bidirectional subsethood based SM [16] [17] with the idea of considering all pairs of continuous subintervals within the DIs. This generalized approach maintains discontinuity information in respect to DIs and uniformly…”

“…The rest of this paper is organized as follows. In Section II, we present some background facts of CIs and DIs, subsethood, two common SMs for the CIs along with the bidirectional subsethood based SM [16] [17]. Section III introduces the proposed generalized SM for the DIs and discusses its properties.…”

“…In this section, we first define CIs and DIs, followed by a review of subsethood, as well as the Jaccard and Dice SMs. Finally, we briefly review the bidirectional subsethood based SM for CIs [16], [17].…”