Nie-Tan Method and its Improved Version: A Counterexample

  • Omar Salazar Universidad Disitrital Francisco Jose de Caldas
  • Juan Diego Rojas Universidad Distrital Francisco Jose de Caldas
  • Humberto Serrano Universidad Distrital Francisco Jose de Caldas
Keywords: Type-2 fuzzy logic system, type-2fuzzy set, centroid, defuzzification, Nie-Tanmethod. (en_US)

Abstract (en_US)

Context: The bottleneck on interval type-2 fuzzy logic systems is the output processing when using Centroid Type-Reduction + Defuzzification (CTR+D method). Nie and Tan proposed an approximation to CTR+D (NT method). Recently, Mendel and Liu improved the NT method (INT method). Numerical examples (due to Mendel and Liu) exhibit the NT and INT methods as good approximations to CTR+D.

Method: Normalization to the unit interval of membership function domains (examples and counterexample) and variables involved in the calculations for the three methods. Examples (due to Mendel and Liu) taken from the literature. Counterexample with piecewise linear membership functions. Comparison by means of error and percentage relative error.

Results: NT vs. CTR+D: Our counterexample showed an error of 0.1014 and a percentage relative error of 30.53%. This is respectively 23 and 32 times higher than the worst case obtained in the examples. INT vs. CTR+D: Our counterexample showed an error of 0.0725 and a percentage relative error of 21.83%. This is respectively 363 and 546 times higher than the worst case obtained in the examples.

Conclusions: NT and INT methods are not necessarily good approximations to the CTR+D method.

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References

J. M. Mendel and R. I. John, “Type-2 fuzzy sets made simple,” IEEE Transactions on Fuzzy Systems, vol. 10, no. 2, pp. 117–127, 2002.

O. Castillo and P. Melin, Type-2 Fuzzy Logic: Theory and Applications, ser. Studies in Fuzziness and Soft Computing. Springer-Verlag Berlin Heidelberg, 2008, vol. 223.

N. N. Karnik and J. M. Mendel, “Type-2 fuzzy logic systems : Type-reduction,” in IEEE International Conference on Systems, Man, and Cybernetics, vol. 2, San Diego, California, USA, Oct. 1998, pp. 2046–2051.

J. M. Mendel, R. I. John, and F. Liu, “Interval type-2 fuzzy logic systems made simple,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 6, pp. 808–821, Dec. 2006.

O. Salazar, J. Soriano, and H. Serrano, “Centroid of an interval type-2 fuzzy set: Continuous vs. discrete,” Ingeniería, vol. 16, no. 2, pp. 67–78, 2011, ISSN 0121-750X.

N. N. Karnik, J. M. Mendel, and Q. Liang, “Type-2 fuzzy logic systems,” IEEE Transactions On Fuzzy Systems, vol. 7, no. 6, pp. 643–658, 1999.

N. N. Karnik and J. M. Mendel, “Centroid of a type-2 fuzzy set,” Information Sciences, vol. 132, pp. 195–220, 2001.

S. Coupland and R. John, “An investigation into alternative methods for the defuzzification of an interval type-2 fuzzy set,” in Proceedings of the 2006 IEEE International Conference on Fuzzy Systems, Vancouver, Canada, Jul. 2006, pp. 1425–1432.

J. M. Mendel, “Type-2 fuzzy sets and systems: An overview,” IEEE Computational Intelligence Magazine, vol. 2, no. 1, pp. 20–29, Feb. 2007.

J. Aisbett, J. T. Rickard, and D. G. Morgenthaler, “Type-2 fuzzy sets as functions on spaces,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 4, pp. 841–844, Aug. 2010.

J.M.Mendel and X. Liu, “New closed-form solutions for karnik-mendel algorithm+defuzzification of an interval type-2 fuzzy set,” in Proceedings of the 2012 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Brisbane, QLD, Jun. 2012, pp. 1–8.

——, “Simplified interval type-2 fuzzy logic systems,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 6, pp. 1056–1069, 2013.

M. Nie and W. W. Tan, “Towards an efficient type-reduction method for interval type-2 fuzzy logic systems,” in Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ 2008), 2008, pp. 1425–1432.

S. Greenfield and F. Chiclana, “Accuracy and complexity evaluation of defuzzification strategies for the discretized interval type-2 fuzzy set,” International Journal of Approximate Reasoning, vol. 54, pp. 1013–1033, 2013.

J. M. Mendel, Uncertainty Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall PTR, 2001.

D. Wu and M. Nie, “Comparison and practical implementation of type-reduction algorithms for type-2 fuzzy sets and systems,” in Proceedings of the IEEE International Conference on Fuzzy Systems (FUZZ 2011), 2011, pp. 2131–2138.

J.M.Mendel and H.Wu, “Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems,”IEEE Transactions on Fuzzy Systems, vol. 10, no. 5, pp. 622–639, 2002.

S. Greenfield, F. Chiclana, S. Coupland, and R. John, “The collapsing method of defuzzification for discretized interval type-2 fuzzy sets,” Information Sciences, vol. 179, no. 13, pp. 2055–2069, 2009.

J. M. Mendel and H. Wu, “Properties of the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule,” in Proceedings of the 2005 International Conference on Fuzzy Systems (FUZZ-IEEE 2005), 2005, pp. 341–346.

——, “New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule,” Information Sciences, vol. 177, pp. 360–377, 2007.

O. Salazar and J. Soriano, “Generating embedded type-1 fuzzy sets by means of convex combination,” in Proceedings of the 2013 IFSA World Congress NAFIPS Annual Meeting, Edmonton, Canada, Jun. 2013, pp. 51–56.

——, “Convex combination and its application to fuzzy sets and interval-valued fuzzy sets I,” Applied Mathematical Sciences, vol. 9, no. 22, pp. 1061–1068, 2015, ISSN 1312-885X.

——, “Convex combination and its application to fuzzy sets and interval-valued fuzzy sets II,” Applied Mathematical Sciences, vol. 9, no. 22, pp. 1069–1076, 2015, ISSN 1312-885X.

J. M. Mendel and F. Liu, “Super-exponential convergence of the Karnik-Mendel algorithms for computing the centroid of an interval type-2 fuzzy set,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 2, pp. 309–320, Apr. 2007.

X. Liu and J. M. Mendel, “Connect karnik-mendel algorithms to root-finding for computing the centroid of an interval type-2 fuzzy set,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 4, pp. 652–665, 2011.

How to Cite
Salazar, O., Rojas, J. D., & Serrano, H. (2016). Nie-Tan Method and its Improved Version: A Counterexample. Ingeniería, 21(2), 138-153. https://doi.org/10.14483/udistrital.jour.reving.2016.2.a02
Published: 2016-05-26
Section
Computational Intelligence