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A Distance Measure Between Fuzzy Variables

  • Juan Carlos Figueroa-GarcíaEmail author
  • Eduyn Ramiro López-Santana
  • Carlos Franco-Franco
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 742)

Abstract

This paper shows some distance measures based on memberships and centroids for comparing fuzzy variables which are commonly used in fuzzy logic systems and rule-based models. An application example is provided, and some interpretation issues are explained.

Notes

Acknowledgments

The authors would like to thank to Prof. Vladik Kreinovich from the Computer Science Dept. of The University of Texas at El Paso - USA, and Prof. Miguel Alberto Melgarejo-Rey from the Eng. Faculty of the Universidad Distrital Francisco José de Caldas - Bogotá, Colombia for their valuable comments and discussions about the topics addressed in this paper.

References

  1. 1.
    Zadeh, L.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8, 199–249 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Upper Saddle River (1992)zbMATHGoogle Scholar
  3. 3.
    Chaudhuri, B., RosenFeld, A.: A modified hausdorff distance between fuzzy sets. Inf. Sci. 118, 159–171 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Nguyen, H.T., Kreinovich, V.: Computing degrees of subsethood and similarity for interval-valued fuzzy sets: fast algorithms. In: 9th International Conference on Intelligent Technologies, InTec 2008, pp. 47–55. IEEE (2008)Google Scholar
  5. 5.
    Zheng, G., Wang, J., Zhou, W., Zhang, Y.: A similarity measure between interval type-2 fuzzy sets. In: IEEE International Conference on Mechatronics and Automation, pp. 191–195. IEEE (2010)Google Scholar
  6. 6.
    Xuechang, L.: Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy Sets Syst. 52, 305–318 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hung, W.L., Yang, M.S.: Similarity measures between type-2 fuzzy sets. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 12(6), 827–841 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Figueroa-García, J.C., Chalco-Cano, Y., Román-Flores, H.: Distance measures for interval type-2 fuzzy numbers. Discrete Appl. Math. 197(1), 93–102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Figueroa-García, J.C., Hernández-Pérez, G.J.: On the computation of the distance between interval type-2 fuzzy numbers using a-cuts. In: IEEE (ed.) Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), vol. 1, pp. 1–6. IEEE (2014)Google Scholar
  10. 10.
    Anderson, D.T., Bezdek, J.C., Popescu, M., Keller, J.M.: Comparing fuzzy, probabilistic, and possibilistic partitions. IEEE Trans. Fuzzy Syst. 18(5), 906–918 (2010)CrossRefGoogle Scholar
  11. 11.
    Hüllermeier, E., Rifqi, M., Henzgen, S., Senge, R.: Comparing fuzzy partitions: a generalization of the Rand index and related measures. IEEE Trans. Fuzzy Syst. 20(3), 546–556 (2012)CrossRefGoogle Scholar
  12. 12.
    Kosko, B.: Fuzziness vs. probability. Int. J. Gen. Syst. 17(1), 211–240 (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ramík, J., Rimánek, J.: Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets Syst. 16, 123–138 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Figueroa-García, J.C.: An approximation method for type reduction of an interval type-2 fuzzy set based on \(\alpha \)-cuts. In: IEEE (ed.) Proceedings of FEDCSIS 2012, pp. 1–6. IEEE (2012)Google Scholar
  15. 15.
    Melgarejo, M., Bernal, H., Duran, K.: Improved iterative algorithm for computing the generalized centroid of an interval type-2 fuzzy set. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), vol. 27, pp. 1–6. IEEE (2008)Google Scholar
  16. 16.
    Melgarejo, M.A.: A fast recursive method to compute the generalized centroid of an interval type-2 fuzzy set. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), pp. 190–194. IEEE (2007)Google Scholar
  17. 17.
    Wu, D., Mendel, J.M.: Enhanced Karnik-Mendel algorithms for interval type-2 fuzzy sets and systems. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), vol. 26, pp. 184–189. IEEE (2007)Google Scholar
  18. 18.
    Wu, D., Mendel, J.M.: Enhanced Karnik-Mendel algorithms. IEEE Trans. Fuzzy Syst. 17(4), 923–934 (2009)CrossRefGoogle Scholar
  19. 19.
    Chalco-Cano, Y., Román-Flores, H.: On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38, 112–119 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chalco-Cano, Y., Román-Flores, H.: Comparation between some approaches to solve fuzzy differential equations. Fuzzy Sets Syst. 160(11), 1517–1527 (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    Figueroa-García, J.C., Neira, D.P.: On ordering words using the centroid and Yager index of an interval type-2 fuzzy number. In: Proceedings of the Workshop on Engineering Applications, WEA 2015, vol. 1, pp. 1–6. IEEE (2015)Google Scholar
  22. 22.
    Figueroa-García, J.C., Neira, D.P.: A comparison between the centroid and the yager index rank for type reduction of an interval type-2 fuzzy number. Revista Ingeniería Universidad Distrital 21(2), 225–234 (2016)Google Scholar
  23. 23.
    Wu, D., Mendel, J.M.: Uncertainty measures for interval type-2 fuzzy sets. Inf. Sci. 177(1), 5378–5393 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Juan Carlos Figueroa-García
    • 1
    Email author
  • Eduyn Ramiro López-Santana
    • 1
  • Carlos Franco-Franco
    • 2
  1. 1.Universidad Distrital Francisco José de CaldasBogotáColombia
  2. 2.Universidad del RosarioBogotáColombia

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