Generation and Reduction of Fuzzy Sets with PG-Means and Fuzzy Similarity Measures

  • Arturo Garcia-GarciaEmail author
  • Andres Mendez-Vazquez
  • Marek Z. Reformat
Part of the Studies in Computational Intelligence book series (SCI, volume 749)


The probabilistic clustering techniques can be applied to generate fuzzy sets in situations where there is little or no information about data. Quite often, they generate a huge number of clusters. These clusters can be interpreted as fuzzy sets in a process of building a fuzzy system. A large number of fuzzy sets introduce noise to the fuzzy system, hence the need to reduce their number. Fuzzy Similarity Measures (FSMs) are widely used for comparison of fuzzy sets. Multiple FSMs have been proposed so far, but identifying a single FSM that is the most suitable for a given task is not always a straightforward process. On many occasions, FSMs are used to reduce a number of fuzzy sets. In this paper, we present the results of analyzing suitability of FSMs to reduce number of fuzzy sets and fuzzy if-then rules for an image segmentation problem. We use a PG-Means algorithm to generate fuzzy sets on both input and output variables. We propose and apply algorithms utilizing FSMs to reduce the number of fuzzy sets and rules. The paper includes a case study investigating the application of the proposed method on two images.


PG-means Fuzzy sets Fuzzy similarity measures Fuzzy sets reduction 



The authors would like to thank prof. Greg Hamerly, from the Baylor University, for letting them use the PG-Means program he developed with Yu Feng [38].


  1. 1.
    C.-F. Juang, C.-Y. Wang, A self-generating fuzzy system with ant and particle swarm cooperative optimization. Expert Syst. Appl. 36, 5362–5370 (2009)CrossRefGoogle Scholar
  2. 2.
    H. Bellaaj, R. Ketata, M. Chtourou, A new method for fuzzy rule base reduction. J. Intell. Fuzzy Syst. 25(3), 605–613 (2013)Google Scholar
  3. 3.
    A. Garcia-Garcia, A. Mendez-Vazquez, Learning fuzzy rules through ant optimization, lasso and dirichlet mixtures, in 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), July 2014, pp. 2558–2565Google Scholar
  4. 4.
    D. Dubois, H. Prade, A unifying view of comparison indices in a fuzzy set-theoretic framework. Fuzzy Sets Possibility Theory: Recent Dev.(Pergamon, New York, 1982)Google Scholar
  5. 5.
    W.-J. Wang, New similarity measures on fuzzy sets and on elements. Fuzzy Sets Syst. 85(3), 305–309 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H. Lee-Kwang, Y.-S. Song, K.-M. Lee, Similarity measure between fuzzy sets and between elements. Fuzzy Sets Syst. 62(3), 291–293 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    X. Wang, B.D. Baets, E. Kerre, A comparative study of similarity measures. Fuzzy Sets Syst. 73(2), 259–268 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    I. Turksen, Z. Zhong, An approximate analogical reasoning schema based on similarity measures and interval-valued fuzzy sets. Fuzzy Sets Syst. 34(3), 323–346 (1990)CrossRefGoogle Scholar
  9. 9.
    S. Santini, R. Jain, Similarity measures. Pattern Anal. Mach. Intell. IEEE Trans. 21(9), 871–883 (1999)CrossRefGoogle Scholar
  10. 10.
    Y. Jin, W. von Seelen, B. Sendhoff, On generating fc3 fuzzy rule systems from data using evolution strategies. Syst. Man Cybern. Part B: Cybern. IEEE Trans. 29(6), 829–845 (1999)CrossRefGoogle Scholar
  11. 11.
    M.-Y. Chen, D. Linkens, Rule-base self-generation and simplification for data-driven fuzzy models. Fuzzy Sets Syst. 142(2), 243–265 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Y. Jin, Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability improvement. Fuzzy Syst. IEEE Trans. 8(2), 212–221 (2000)CrossRefGoogle Scholar
  13. 13.
    T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control. Syst. Man Cybern. IEEE Trans. SMC-15(1), 116–132 (1985)Google Scholar
  14. 14.
    L.A. Zadeh, Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefzbMATHGoogle Scholar
  15. 15.
    C. Cornelis, E. Kerre, Inclusion Measures in Intuitionistic Fuzzy Set Theory. (Berlin, Heidelberg: Springer Berlin Heidelberg, 2003), pp. 345–356. [Online]. Available:
  16. 16.
    G. Bordogna, P. Bosc, G. Pasi, Fuzzy inclusion in database and information retrieval query interpretation, in Proceedings of the 1996 ACM Symposium on Applied Computing, ser. SAC’96. New York, NY, USA: ACM, 1996, pp. 547–551. [Online]. Available:
  17. 17.
    P. Bosc, O. Pivert, On a reinforced fuzzy inclusion and its application to database querying. (Berlin, Heidelberg: Springer Berlin Heidelberg, 2012), pp. 351–360. [Online]. Available:
  18. 18.
    D. Sinha, E.R. Dougherty, Fuzzification of set inclusion: theory and applications. Fuzzy Sets Syst. 55(1), 15–42 (1993). [Online]. Available:
  19. 19.
    C. Cornelis, C. V. der Donck, E. Kerre, Sinhadougherty approach to the fuzzification of set inclusion revisited. Fuzzy Sets Syst, 134(2), 283–295 (2003). [Online]. Available:
  20. 20.
    I. Beg, S. Ashraf, Fuzzy inclusion and design of measure of fuzzy inclusion. RIMAI J., 8 (2012)Google Scholar
  21. 21.
    M. Wygralak, Fuzzy inclusion and fuzzy equality of two fuzzy subsets, fuzzy operations for fuzzy subsets, Fuzzy Sets Syst. 10(13), 157–168 (1983). [Online].
  22. 22.
    A. Pal, B. Mondal, N. Bhattacharyya, S. Raha, Similarity in fuzzy systems. J. Uncertainty Anal. Appl. 2(1) (2014)Google Scholar
  23. 23.
    C.P. Pappis, N.I. Karacapilidis, A comparative assessment of measures of similarity of fuzzy values. Fuzzy Sets Syst. 56(2), 171–174 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    W. Zeng, H. Li, Inclusion measures, similarity measures, and the fuzziness of fuzzy sets and their relations. Int. J. Intell. Syst. 21(6), 639–653 (2006)CrossRefzbMATHGoogle Scholar
  25. 25.
    T. Liao, Z. Zhang, A review of similarity measures for fuzzy systems, in Fuzzy Systems, 1996. Proceedings of the Fifth IEEE International Conference on, vol. 2, Sep 1996, pp. 930–935Google Scholar
  26. 26.
    M. Alamuri, B. Surampudi, A. Negi, A survey of distance/similarity measures for categorical data, in Neural Networks (IJCNN), 2014 International Joint Conference on, July 2014, pp. 1907–1914Google Scholar
  27. 27.
    S. Raha, N. Pal, K. Ray, Similarity-based approximate reasoning: methodology and application. Syst. Man Cybern. Part A: Syst. Humans IEEE Trans. 32(4), 541–547 (2002)CrossRefGoogle Scholar
  28. 28.
    H. L. Capitaine, C. Fr ́elicot, Towards a unified logical framework of fuzzy implications to compare fuzzy sets, in Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, July 20–24, 2009, pp. 1200–1205Google Scholar
  29. 29.
    V.R. Young, Fuzzy subsethood. Fuzzy Sets Syst. 77(3), 371–384 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Goguen, The logic of inexact concepts. Synthese 19(3–4), 325–373 (1969). [Online]. Available:
  31. 31.
    D. Dubois, H. Prade, Fuzzy sets and systems—theory and applications (Academic press, New York, 1980)zbMATHGoogle Scholar
  32. 32.
    M. Wygralak, Fuzzy inclusion and fuzzy equality of two fuzzy subsets, fuzzy operations for fuzzy subsets. Fuzzy Sets Syst. 10(1), 157–168 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    M. Nachtegael, H. Heijmans, D. Van der Weken, E. Kerre, Fuzzy adjunctions in mathematical morphology, in Proceedings of JCIS, 2003, pp. 202–205Google Scholar
  34. 34.
    P. Bosc, A. Hadjali, O. Pivert, Graded tolerant inclusion and its axiomatization, in Proceedings of the 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), 2008Google Scholar
  35. 35.
    I. Beg, S. Ashraf, Fuzzy relational calculus. Bull. Malays. Math. Sci. Soc. 2 (2013)Google Scholar
  36. 36.
    I. Beg, S. Ashraf, Kleene’s fuzzy similarity and measure of similarity. Ann. Fuzzy Math. Inform. 6(2), 251–261 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    A. Garcia-Garcia, M. Reformat, A. Mendez-Vazquez, Similarity-based method for reduction of fuzzy rules, in 2014 North American Fuzzy Information Processing Society (NAFIPS), October 2016Google Scholar
  38. 38.
    Y.F.G. Hamerly, PG-means: learning the number of clusters in data. Adv. Neural. Inf. Process. Syst. 19, 393–400 (2007)Google Scholar
  39. 39.
    A. Rosenberg, Automatic detection and classification of prosodic events, Ph.D. Dissertation, Columbia University, 2009Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Arturo Garcia-Garcia
    • 1
    Email author
  • Andres Mendez-Vazquez
    • 1
  • Marek Z. Reformat
    • 2
  1. 1.CINVESTAV, Computer ScienceZapopanMexico
  2. 2.University of AlbertaEdmontonCanada

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