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Type-2 Fuzzy Sets

  • Jerry M. Mendel
Chapter
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Abstract

This chapter formally introduces type-2 fuzzy sets and is the backbone for the rest of this book. It includes a lot of new terminologies. Coverage includes: the concept of a type-2 fuzzy set, definitions of general type-2 fuzzy sets and associated concepts, definitions of interval type-2 fuzzy sets and associated concepts, examples of two popular footprints of uncertainty, interval type-2 fuzzy numbers, a hierarchy of different kinds of type-2 fuzzy sets, mathematical representations of type-2 fuzzy sets including the vertical slice, wavy slice, and horizontal slice representations, which mathematical representations are most useful for optimal design applications, how to represent non-type-2 fuzzy sets as type-2 fuzzy sets, returning to linguistic labels for type-2 fuzzy sets, and multivariable MFs. 24 examples are used to illustrate the important concepts.

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References

  1. Aisbett, J., J.T. Rickard, and D.G. Morgenthaler. 2010. Type-2 fuzzy sets as functions on spaces. IEEE Transactions on Fuzzy Systems 18: 841–844.CrossRefGoogle 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.CrossRefGoogle Scholar
  3. Bilgin, A., H. Hagras, A. Malibari, M. Alhaddad, and D. Alghazzawi. 2012a. Towards a general type-2 fuzzy logic approach for computing with words using linear adjectives. In Proceedings FUZZ-IEEE 2012, pp. 1130–1137, Brisbane, AU.Google Scholar
  4. Bilgin, A., J.Dooley, L. Whittington, H. Hagras, M. Henson, C. Wagner, A. Malibari, A. Al-Ghamdi, M. Alhaddad, and D. Alghazzawi. 2012b. Dynamic profile-selection for zslices based type-2 fuzzy agents controlling multi-user ambient intelligent environments. In Proceedings FUZZ-IEEE 2012, pp. 1392–1399, Brisbane, AU.Google Scholar
  5. Bilgin, A., H. Hagras, A. Malibari, M. J. Alhaddad, and D. Alghazzawi. 2012c. A general type-2 fuzzy logic approach for adaptive modeling of perceptions for computing with words. In Proceedings 2012 12th UK workshop on computational intelligence (UKCI), pp. 1–8.Google Scholar
  6. Bilgin, A., H. Hagras, A. Malibari, M. J. Alhaddad, and D. Alghazzawi. 2013a. An experience based linear general type-2 fuzzy logic approach for computing with words. In Proceedings of IEEE international conference on fuzzy systems, Paper #1139, Hyderabad, India.Google Scholar
  7. Bilgin, A., H. Hagras, A. Malibari, M. J. Alhaddad, and D. Alghazzawi. 2013b. Towards a linear general type-2 fuzzy logic based approach for computing with words. International Journal of Soft Computing 17:2203–2222.Google Scholar
  8. Buckley, J.J. 2003. Fuzzy probabilities: New approaches and new applications. New York: Physica-Verlag.CrossRefzbMATHGoogle Scholar
  9. Bustince, H. 2000. Indicator of inclusion grade for interval-valued fuzzy sets: Applications to approximate reasoning based on interval-valued fuzzy sets. International Journal of Approximate Reasoning 23: 137–209.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Bustince, H., J. Fernandez, H. Hagras, F. Herrera, M. Pagola, and E. Barrenechea. 2015. Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: towards a wider view on their relationship. IEEE Transactions on Fuzzy Systems 23: 1876–1882.CrossRefGoogle Scholar
  11. Chen, Q., and S. Kawase. 2000. On fuzzy-valued fuzzy reasoning. Fuzzy Sets and Systems 113: 237–251.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Coupland, S., and R.I. John. 2007. Geometric type-1 and type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems 15: 3–15.CrossRefzbMATHGoogle Scholar
  13. Garibaldi, J. M., S. Musikasuwan, and T. Ozen. 2005. The association between non-stationary and interval type-2 fuzzy sets: A case study. In Proceedings IEEE FUZZ conference, pp. 224–229, Reno, NV.Google Scholar
  14. Gorzalczany, M.B. 1987. A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems 21: 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Greenfield, S., and R. John. 2007. Optimized generalized type-2 join and meet operations. In Proceedings FUZZ-IEEE 2007, pp. 141–146, London, UK.Google Scholar
  16. Hamrawi, H., and S. Coupland. 2009. Type-2 fuzzy arithmetic using alpha-planes. In Proceedings IFSA/EUSFLAT, pp. 606–611, Portugal.Google Scholar
  17. Hamrawi, H., S. Coupland, and R. John. 2010. A novel alpha-cut representation for type-2 fuzzy sets. In Proceedings FUZZ-IEEE 2010, IEEE world congress on computational intelligence, pp. 351–358, Barcelona, Spain.Google Scholar
  18. John, R., and S. Coupland. 2012. type-2 fuzzy logic: Challenges and misconceptions. IEEE Computational Intelligence Magazine 7 (3): 48–52.CrossRefGoogle Scholar
  19. Karnik, N.N., and J.M. Mendel. 1998. An introduction to type-2 fuzzy logic systems. USC-SIPI Report #418, Univ. of Southern Calif., Los Angeles, CA, June 1998. This can be accessed at: http://sipi.usc.edu/research; then choose “sipi technical reports/418.”.
  20. Karnik, N.N., and J.M. Mendel. 2001a. Operations on type-2 fuzzy sets. Fuzzy Sets and Systems 122: 327–348.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Karnik, N.N., and J.M. Mendel. 2001b. Centroid of a type-2 fuzzy set. Information Sciences 132: 195–220.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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
  23. Klir, G.J., and B. Yuan. 1995. Fuzzy sets and fuzzy logic: Theory and applications. Upper Saddle River, NJ: Prentice Hall.zbMATHGoogle Scholar
  24. Kumbasar, T., and H. Hagras. 2015. A self-tuning zSlices based general type-2 fuzzy PI controller. IEEE Transactions on Fuzzy Systems 23: 991–1013.CrossRefGoogle Scholar
  25. Liang, Q., and J.M. Mendel. 2000. Interval type-2 fuzzy logic systems. In Proceedings FUZZ-IEEE ’00, San Antonio, TX.Google Scholar
  26. Liu, F. 2008. An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Information Sciences 178: 2224–2236.MathSciNetCrossRefGoogle Scholar
  27. Liu, F., and J.M. Mendel. 2008. Encoding words into interval type-2 fuzzy sets using an interval approach. IEEE Transactions on Fuzzy Systems 16: 1503–1521.CrossRefGoogle Scholar
  28. Ljung, L. 1999. System identification: Theory for the user, 2nd ed. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar
  29. Lushu, L. 1995. Random fuzzy sets and fuzzy martingales. Fuzzy Sets and Systems 69: 181–192.MathSciNetCrossRefzbMATHGoogle Scholar
  30. McCulloch, J., and C. Wagner. 2016. Measuring the similarity between zSlices general type-2 fuzzy sets with non-normal secondary membership functions. In Proceedings of FUZZ-IEEE 2016, pp. 461–468, Vancouver, Canada.Google Scholar
  31. Mendel, J.M. 2001. Introduction to rule-based fuzzy logic systems. Upper Saddle River, NJ: Prentice-Hall.zbMATHGoogle Scholar
  32. Mendel, J.M. 2007. Type-2 fuzzy sets and systems: An overview. IEEE Computational Intelligence Magazine 2: 20–29.Google Scholar
  33. Mendel, J.M. 2009. On answering the question ‘Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?’. Information Sciences 179: 3418–3431.MathSciNetCrossRefzbMATHGoogle Scholar
  34. Mendel, J.M. 2010. Comments on ‘α-plane representation for type-2 fuzzy sets: Theory and applications’. IEEE Transactions on Fuzzy Systems 18: 229–230.CrossRefGoogle Scholar
  35. Mendel, J.M. 2012. Plotting 2-1/2 D figures for general type-2 fuzzy sets by hand or by powerpoint. In Proceedings FUZZ-IEEE 2012, pp. 1490-1497, Brisbane, AU.Google Scholar
  36. Mendel, J.M. 2014. General type-2 fuzzy logic systems made simple: A tutorial. IEEE Transactions on Fuzzy Systems 22: 1162–1182.CrossRefGoogle Scholar
  37. Mendel, J.M., and R.I. John. 2002. Type-2 fuzzy sets made simple. IEEE Transactions on Fuzzy Systems 10: 117–127.CrossRefGoogle Scholar
  38. Mendel, J.M., and Q. Liang. 1999. Pictorial comparisons of type-1 and type-2 fuzzy logic systems. In Proceedings IASTED international conference on intelligent systems & control, Santa Barbara, CA.Google Scholar
  39. Mendel, J.M., and D. Wu. 2010. Perceptual computing: Aiding people in making subjective judgments. Hoboken, NJ: Wiley and IEEE Press.CrossRefGoogle Scholar
  40. Mendel, J.M., R.I. John, and F. Liu. 2006. Interval type-2 fuzzy logic systems made simple. IEEE Transactions on Fuzzy Systems 14: 808–821.CrossRefGoogle Scholar
  41. Mendel, J.M., F. Liu, and D. Ƶhai. 2009. Alpha-plane representation for type-2 fuzzy sets: Theory and applications. IEEE Transactions an Fuzzy Systems 17: 1189–1207.Google Scholar
  42. Mendel, J.M., M.R. Rajati, and P. Sussner. 2016. On clarifying some notations used for type-2 fuzzy sets as well as some recommended notational changes. Information Sciences 340–341: 337–345.CrossRefGoogle Scholar
  43. Mizumoto, M., and K. Tanaka. 1976. Some properties of fuzzy sets of type-2. Information and Control 31: 312–340.MathSciNetCrossRefzbMATHGoogle Scholar
  44. Mizumoto, M., and K. Tanaka. 1981. Fuzzy sets of type-2 under algebraic product and algebraic sum. Fuzzy Sets and Systems 5: 277–290.MathSciNetCrossRefzbMATHGoogle Scholar
  45. Moharrer, M., H. Tahayori, H., and A. Sadeghian. 2013. Modeling complex concepts with type-2 fuzzy sets: The case of user satisfaction of online services. In Advances in type-2 fuzzy sets and systems: theory and applications, ed. Sadeghian, et al., 133–146. NY: Springer.Google Scholar
  46. Moller, B., and M. Beere. 2004. Fuzzy randomness: Uncertainty in civil engineering and computational mechanics. New York: Springer.CrossRefGoogle Scholar
  47. Pedrycz, W. 2015. Concepts and design aspects of granular models of type-1 and type-2. International Journal of Fuzzy Logic and Intelligent Systems 15: 87–95.Google Scholar
  48. Rakshit, P., A. Chakraborty, A. Konar, and A.K. Nagar. 2013. Secondary membership evaluation in generalized type-2 fuzzy sets by evolutionary optimization algorithm. In Proceedings FUZZ-IEEE 2013, Paper #1334, Hyderabad, India.Google Scholar
  49. Rakshit, P., A. Saha, A. Konar, and S. Saha. 2016. A type-2 fuzzy classifier for gesture recognition induced pathological disorder recognition. Fuzzy Sets and Systems 305: 95–130.Google Scholar
  50. Starczewski, J.T. 2009a. Efficient triangular type-2 fuzzy logic systems. International Journal of Approximate Reasoning 50: 799–811.CrossRefzbMATHGoogle Scholar
  51. Starczewski, J.T. 2009b. Extended triangular norms. Information Sciences 179: 742–757.MathSciNetCrossRefzbMATHGoogle Scholar
  52. Tahayori, H., A.G.B. Tettamanzi, and G.D. Antoni. 2006. Approximated type-2 fuzzy set operations. In Proceedings FUZZ-IEEE 2006, Vancouver, B. C. Canada, pp. 9042–9049.Google Scholar
  53. Tahayori, H., A.G.B. Tettamanzi, G.D. Antoni, A. Visconti, and M. Moharrer. 2010. Concave type-2 fuzzy sets: Properties and operations. Soft Computing Journal 14(7): 749–756.CrossRefzbMATHGoogle Scholar
  54. Ulu, C., M. Güzellkaya, and I. Eksin. 2013. Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets. Applied Soft Computing 13: 3713–3728.CrossRefGoogle Scholar
  55. Wagner, C., and H. Hagras. 2008. z Slices–towards bridging the gap between interval and general type-2 fuzzy logic. In Proceedings IEEE FUZZ conference, Paper # FS0126, Hong Kong, China.Google Scholar
  56. Wagner, C., and H. Hagras. 2010. Towards general type-2 fuzzy logic systems based on zSlices. IEEE Transactions on Fuzzy Systems 18: 637–660.CrossRefGoogle Scholar
  57. Wagner, C., and H. Hagras. 2013. zSlices based general type-2 fuzzy sets and systems. In Advances in type-2 fuzzy sets and systems: Theory and applications, eds. Sadeghian, A., J.M. Mendel, and H. Tahayori. New York: Springer.Google Scholar
  58. Wu, D. 2011. A constrained representation theorem for interval type-2 fuzzy sets using convex and normal embedded type-1 fuzzy sets and its application to centroid computation. In Proceedings of world conference on soft computing, Paper #200, San Francisco, CA.Google Scholar
  59. Wu, D., and J.M. Mendel. 2007. Uncertainty measures for interval type-2 fuzzy sets. Information Sciences 177: 5378–5393.MathSciNetCrossRefzbMATHGoogle Scholar
  60. Zadeh, L.A. 1975. The concept of a linguistic variable and its application to approximate reasoning–1. Information Sciences 8: 199–249.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA

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