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Type-Reduction

  • Jerry M. Mendel
Chapter

Abstract

This chapter introduces a computation called type-reduction that lets a type-2 fuzzy set (T2 FS) be projected into a type-1 fuzzy set. Type-reduction is often used in a type-2 fuzzy system as a first step in going from a T2 FS to a number. Coverage includes: the interval weighted average (IWA), because it is the basic building block for type-reduction; three algorithms (KM, EKM, and EIASC) for computing the IWA; centroid type-reduction for interval T2 FSs and systems; height and center-of-sets type-reduction for IT2 fuzzy systems; centroid type-reduction for general T2 FSs and systems; height and center-of-sets type-reduction for GT2 fuzzy systems; an appendix that presents (for historical reasons) the early approach to type-reduction; and, an appendix about the mathematical properties of the IWA, and about continuous algorithms for performing centroid type-reduction. Fourteen examples are used to illustrate the important concepts.

References

  1. Alefeld, G. 1981. On the convergence of Halley’s method. American Mathematical Monthly 88 (7): 530–536.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Almaraashi, M., R. John, A. Hopgood, and S. Ahmadi 2016. Learning of interval and general type-2 fuzzy logic systems using simulated annealing: Theory and practice. Information Sciences 360: 21–42.Google Scholar
  3. Chen, C.-L., S.-C. Chen, and Y.-H. Kuo. 2014. The reduction of interval type-2 LR fuzzy sets. IEEE Transactions on Fuzzy Systems 22: 840–858.CrossRefGoogle Scholar
  4. Cover, T.M., and J.A. Thomas. 1991. Elements of information theory. New York: Wiley.CrossRefzbMATHGoogle Scholar
  5. Duran, K., H. Bernal, and M. Melgarejo. 2008. Improved iterative algorithm for computing the generalized centroid of an interval type-2 fuzzy set. NAFIPS 2008, Paper 50056, New York City, May 2008.Google Scholar
  6. Greenfield, S., R.I. John, and S. Coupland. 2005. A novel sampling method for type-2 defuzzification. In Proceedings of UKCI 2005, 120–127, London, September 2005.Google Scholar
  7. Han, S., and X. Liu. 2016. Global convergence of Karnik-Mendel algorithms. Fuzzy Sets and Systems.Google Scholar
  8. Hu, H., Y. Wang, and Y. Cai. 2012a. Advantages of the enhanced opposite direction searching algorithm for computing the centroid of an interval type-2 fuzzy set. Asian Journal of Control 14 (6): 1–9.MathSciNetzbMATHGoogle Scholar
  9. Hu, H., G. Zhao, and H.N. Yang. 2012b. Fast algorithm to calculate generalized centroid of interval type-2 fuzzy set. Control and Decision 25 (4): 637–640.MathSciNetGoogle Scholar
  10. John, R.I. 2000. Perception modelling using type-2 fuzzy sets. Ph. D. thesis, De Montfort University.Google Scholar
  11. Karnik, N.N., and J.M. Mendel. 1998. An introduction to type-2 fuzzy logic systems. USC-SIPI Report #418, University of Southern California, Los Angeles, CA, June 1998. This can be accessed at: http://sipi.usc.edu/research; then choose sipi technical reports/418.
  12. Karnik, N.N., and J.M. Mendel. 2001. Centroid of a type-2 fuzzy set. Information Sciences 132: 195–220.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Karnik, N.N., J.M. Mendel, and Q. Liang. 1999. Type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems 7: 643–658.CrossRefGoogle Scholar
  14. Klir, G.J., and T.A. Folger. 1988. Fuzzy sets, uncertainty, and information. Englewood Cliffs, NJ: Prentice Hall.zbMATHGoogle Scholar
  15. Li, C., J. Yi, and D. Zhao. 2008. A novel type-reduction method for interval type-2 fuzzy logic systems. In: Proceedings 5th international conference on fuzzy systems knowledge discovery, vol. 1, 157–161, Jinan, China.Google Scholar
  16. Linda, O., and M. Manic. 2012. Monotone centroid flow algorithm for type reduction of general type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems 20: 805–819.CrossRefGoogle Scholar
  17. Liu, F. 2008. An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Information Sciences 178: 2224–2236.MathSciNetCrossRefGoogle Scholar
  18. Liu, X., Y. Qin, and L. Wu. 2012b. Fast and direct Karnik-Mendel algorithm computation for the centroid of an interval type-2 fuzzy set. In: Proceedings of FUZZ-IEEE 2012, 1058–1065, Brisbane, AU, June 2012.Google Scholar
  19. Liu, F., and J.M. Mendel. 2008. Aggregation using the fuzzy weighted average, as computed by the KM algorithms. IEEE Transactions on Fuzzy Systems 16: 1–12.CrossRefGoogle Scholar
  20. Liu, X., and J.M. Mendel. 2011. Connect Karnik-Mendel algorithms to root-finding for computing the centroid of an interval type-2 fuzzy set. IEEE Transactions on Fuzzy Systems 19: 652–665.CrossRefGoogle Scholar
  21. Liu, X., J.M. Mendel, and D. Wu. 2012a. Study on enhanced Karnik-Mendel algorithms: Initialization explanations and computation improvements. Information Sciences 187: 75–91.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lucas, L.A., T.M. Centeno, and R.M. Delgado. 2007. General type-2 inference systems: Analysis, design and computational aspects. In Proceedings of FUZZ-IEEE-2007, London, UK, 1107–1112.Google Scholar
  23. Melgarejo, M.C.A. 2007. A fast recursive method to compute the generalized centroid of an interval type-2 fuzzy set. In Proceedings of North American fuzzy information processing society (NAFIPS), 190–194.Google Scholar
  24. Mendel, J.M. 2001. Introduction to rule-based fuzzy logic systems. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar
  25. Mendel, J.M. 2005. On a 50% savings in the computation of the centroid of a symmetrical interval type-2 fuzzy set. Information Sciences 172: 417–430.MathSciNetCrossRefGoogle Scholar
  26. Mendel, J.M. 2007. Advances in type-2 fuzzy sets and systems. Information Sciences 177: 84–110.MathSciNetCrossRefzbMATHGoogle Scholar
  27. Mendel, J.M. 2013. On KM algorithms for solving type-2 fuzzy set problems. IEEE Transactions on Fuzzy Systems 21: 426–446.CrossRefGoogle Scholar
  28. Mendel, J.M. 2014. General type-2 fuzzy logic systems made simple: A tutorial. IEEE Transactions on Fuzzy Systems 22: 1162–1182.CrossRefGoogle Scholar
  29. Mendel, J.M. 2015. Type-2 fuzzy sets and systems: A retrospective. Informatik Spektrum 38 (6): 523–532.CrossRefGoogle Scholar
  30. Mendel, J.M., and R.I. John. 2002. Type-2 fuzzy sets made simple. IEEE Transactions on Fuzzy Systems 10: 117–127.CrossRefGoogle Scholar
  31. Mendel, J.M., and H. Wu. 2006. Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 1, forward problems. IEEE Transactions on Fuzzy Systems 14: 781–792.CrossRefGoogle Scholar
  32. Mendel, J.M., and H. Wu. 2007. New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule. Information Sciences 177: 360–377.MathSciNetCrossRefzbMATHGoogle Scholar
  33. Mendel, J.M., and D. Wu. 2010. Perceptual computing: Aiding people in making subjective judgments. Hoboken, NJ: Wiley and IEEE Press.CrossRefGoogle Scholar
  34. Mendel, J.M., F. Liu, and D. Zhai. 2009. Alpha-plane representation for type-2 fuzzy sets: Theory and applications. IEEE Transactions on Fuzzy Systems 17: 1189–1207.CrossRefGoogle Scholar
  35. Mendel, J.M., H. Hagras, W.-W. Tan, W.W. Melek, and H. Ying. 2014. Introduction to type-2 fuzzy logic control. Hoboken, NJ: Wiley and IEEE Press.CrossRefzbMATHGoogle Scholar
  36. Nie, M., and W.W. Tan. 2008. Towards an efficient type-reduction method for interval type-2 fuzzy logic systems. In Proceedings of IEEE FUZZ conference, Paper # FS0339, Hong Kong, China, June 2008.Google Scholar
  37. Niewiadomski, A., J. Ochelska, and P.S. Szczepaniak. 2006. Interval-valued linguistic summaries of databases. Control & Cybernetics 35 (2): 415–443.zbMATHGoogle Scholar
  38. Salaken, S.M., A. Khosravi, and S. Nahavandi. 2016. Modification on enhanced Karnik-Mendel algorithm. Expert Systems With Applications.Google Scholar
  39. Starczewski, J.T. 2009. Efficient triangular type-2 fuzzy logic systems. International Journal of Approximate Reasoning 50: 799–811.CrossRefzbMATHGoogle Scholar
  40. Tjalling, J.Y. 1995. Historical development of the Newton-Raphson method. SIAM Review 37 (4): 531–551.MathSciNetCrossRefzbMATHGoogle Scholar
  41. Ulu, C., M. Güzellkaya, and I. Eksin. 2013. A closed form type reduction method for piecewise linear interval type-2 fuzzy sets. International Journal of Approximate Reasoning 54: 1421–1433.MathSciNetCrossRefzbMATHGoogle Scholar
  42. Wu, D. 2011. An interval type-2 fuzzy logic system cannot be implemented by traditional type-1 fuzzy logic systems. In World conference on soft computing, San Francisco, CA, May 2011.Google Scholar
  43. Wu, D. 2012. On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers. IEEE Transactions on Fuzzy Systems 20: 832–848.Google Scholar
  44. Wu, D. 2013. Approaches for reducing the computational costs of interval type-2 fuzzy logic systems: Overview and comparisons. IEEE Transactions on Fuzzy Systems 21 (1): 80–93.CrossRefGoogle Scholar
  45. Wu D., and M. Nie. 2011. Comparison and practical implementations of type-reduction algorithms for type-2 fuzzy sets and systems. In Proceedings of FUZZ-IEEE 2011, 2131–2138, Taipei, Taiwan, June 2011.Google Scholar
  46. Wu, H., and J.M. Mendel. 2002. Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems 10: 622–639.CrossRefGoogle Scholar
  47. Wu, D., and J.M. Mendel. 2007a. Aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems 15 (6): 1145–1161.CrossRefGoogle Scholar
  48. Wu, H., and J.M. Mendel. 2007b. Classification of battlefield ground vehicles using acoustic features and fuzzy logic rule-based classifiers. IEEE Transactions on Fuzzy Systems 15: 56–72.CrossRefGoogle Scholar
  49. Wu, D., and J.M. Mendel. 2009. Enhanced Karnik-Mendel algorithms. IEEE Transactions on Fuzzy Systems 17: 923–934.CrossRefGoogle Scholar
  50. Wu, H.-J., Y.-L. Su, and S.-J. Lee. 2012. A fast method for computing the centroid of a type-2 fuzzy set. IEEE Transactions on Systems, Man, and Cybernetics-Part B (Cybernetics) 42: 764–777.CrossRefGoogle Scholar
  51. Xie, B.-K., and S.-J. Lee. 2016. An extended type-reduction method for general type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems, accepted for publication, March 2016.Google Scholar
  52. Yeh, C.-Y., W.-H. Roger Jeng, and S.-J. Lee. 2011. An enhanced type-reduction algorithm for type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems 19: 227–240.Google Scholar
  53. Zhai, D., and J.M. Mendel. 2011. Computing the centroid of a general type-2 fuzzy set by means of the centroid flow algorithm. IEEE Transactions on Fuzzy Systems 19: 401–422.CrossRefGoogle Scholar
  54. Zhai, D., and J.M. Mendel. 2012. Enhanced centroid-flow algorithm for computing the centroid of general type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems 20: 939–956.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

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