A Direct Approach for Determining the Switch Points in the Karnik–Mendel Algorithm

Chen, Chao; John, Robert; Twycross, Jamie; Garibaldi, Jonathan M.

Abstract

The Karnik-Mendel algorithm is used to compute the centroid of interval type-2 fuzzy sets, determining the switch points needed for the lower and upper bounds of the centroid, through an iterative process. It is commonly acknowledged that there is no closed-form solution for determining such switch points. Many enhanced algorithms have been proposed to improve the computational efficiency of the Karnik-Mendel algorithm. However, all of these algorithms are still based on iterative procedures. In this paper, a direct approach based on derivatives for determining the switch points without multiple iterations has been proposed, together with mathematical proof that these switch points are correctly determining the lower and upper bounds of the centroid. Experimental simulations show that the direct approach obtains the same switch points, but is more computationally efficient than any of the existing (iterative) algorithms. Thus, we propose that this algorithm should be used in any application of interval type-2 fuzzy sets in which the centroid is required.

Citation

Chen, C., John, R., Twycross, J., & Garibaldi, J. M. (2018). A Direct Approach for Determining the Switch Points in the Karnik–Mendel Algorithm. IEEE Transactions on Fuzzy Systems, 26(2), 1079-1085. https://doi.org/10.1109/tfuzz.2017.2699168

• Journal Article Type: Article
• Acceptance Date: Apr 15, 2017
• Online Publication Date: Apr 27, 2017
• Publication Date: 2018-04
• Deposit Date: Apr 20, 2017
• Publicly Available Date: Apr 27, 2017
• 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: 26
• Issue: 2
• Pages: 1079-1085
• DOI: https://doi.org/10.1109/tfuzz.2017.2699168
• Keywords: Karnik-Mendel algorithm, centroid, interval type-2, fuzzy sets, iterative, closed-form, direct approach
• Public URL: https://nottingham-repository.worktribe.com/output/929379
• Publisher URL: http://ieeexplore.ieee.org/document/7913651/
• Additional Information: (c) 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, 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 components of this work in other works.
• Contract Date: Apr 20, 2017