Measuring the directional or non-directional distance between type-1 and type-2 fuzzy sets with complex membership functions

McCulloch, Josie; Wagner, Christian

Abstract

Fuzzy sets may have complex, non-normal or non-convex membership functions that occur, for example, in the output of a fuzzy logic system or when automatically generating fuzzy sets from data. Measuring the distance between such non-standard fuzzy sets can be challenging as there is no clear correct method of comparison and limited research currently exists that systematically compares existing distance measures for these fuzzy sets. It is useful to know the distance between these sets, which can tell us how much the results of a system change when the inputs differ, or the amount of disagreement between individual's perceptions or opinions on different concepts. In addition, understanding the direction of difference between such fuzzy sets further enables us to rank them, learning if one represents a higher output or higher ratings than another. This paper picks up previous functions of measuring directional distance and, for the first time, presents methods of measuring the directional distance between any type-1 and type-2 fuzzy sets with both normal/non-normal and convex/non-convex membership functions. In real-world applications where data-driven, non-convex, non-normal fuzzy sets are the norm, the proposed approaches for measuring the distance enables us to systematically reason about the real-world objects captured by the fuzzy sets.

Citation

McCulloch, J., & Wagner, C. (2019). Measuring the directional or non-directional distance between type-1 and type-2 fuzzy sets with complex membership functions. IEEE Transactions on Fuzzy Systems, 27(7), 1506-1515. https://doi.org/10.1109/tfuzz.2018.2882342

• Journal Article Type: Article
• Acceptance Date: Nov 15, 2018
• Online Publication Date: Nov 22, 2018
• Publication Date: 2019-07
• Deposit Date: Nov 26, 2018
• Publicly Available Date: Nov 26, 2018
• 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: 27
• Issue: 7
• Pages: 1506-1515
• DOI: https://doi.org/10.1109/tfuzz.2018.2882342
• Public URL: https://nottingham-repository.worktribe.com/output/1306469
• Publisher URL: https://ieeexplore.ieee.org/document/8543161
• Additional Information: © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.