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A Survey of Continuous Karnik–Mendel Algorithms and Their Generalizations

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Advances in Type-2 Fuzzy Sets and Systems

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 301))

Abstract

Karnik–Mendel (KM) algorithms are important tools for type-2 fuzzy logic. This survey chapter summarizes some extensions of continuous Karnik–Mendel algorithms. It is shown that the solution of KM algorithms can be transformed into the solution of root-finding problems, and that the iteration formula in KM algorithms is equivalent to the Newton-Raphson root-finding method in numerical analysis. New iteration formulas are summarized that accelerate the convergence speed and it is shown that numerical integration methods can be used to improve computation accuracy. This chapter demonstrates that properties and structures of KM algorithms can be understood and improved with the techniques from numerical analysis.

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Notes

  1. 1.

    As noted in [25, p. 363], if Gaussian MFs are used, one can extend the theoretical results to \(a\rightarrow -\infty \), \(b\rightarrow +\infty \); but, in practice, when truncations are used, \(a\) and \(b\) are again finite numbers.

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Acknowledgments

This work has been supported by the Natural Science Foundation of China Project under Grant Nos. 70771025 and 701171048. Part of the work in this chapter was done jointly with Professor Jerry M. Mendel and Dr. Dongrui Wu in [10, 11]. I would like to thank Professor Jerry M. Mendel and anonymous reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Economics and Management, Southeast University, Nanjing, 210096, People’s Republic of China

    Xinwang Liu

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  1. Xinwang Liu
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Correspondence to Xinwang Liu .

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Editors and Affiliations

  1. Ryerson University, Victoria Street 350, Toronto, M5B 2K3, Ontario, Canada

    Alireza Sadeghian

  2. Hughes Aircraft Electrical Engineering B, Department of Electrical Engineering - S, University of Southern California, 3740 McClintock Ave., Los Angeles, 90089-2564, California, USA

    Jerry M. Mendel

  3. Ryerson University, Victoria Street 350, Toronto, M5B 2K3, Ontario, Canada

    Hooman Tahayori

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Liu, X. (2013). A Survey of Continuous Karnik–Mendel Algorithms and Their Generalizations. In: Sadeghian, A., Mendel, J., Tahayori, H. (eds) Advances in Type-2 Fuzzy Sets and Systems. Studies in Fuzziness and Soft Computing, vol 301. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6666-6_2

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  • DOI: https://doi.org/10.1007/978-1-4614-6666-6_2

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-6665-9

  • Online ISBN: 978-1-4614-6666-6

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