Direct Factorization by Similarity of Fuzzy Concept Lattices by Factorization of Input Data

  • Radim Belohlavek
  • Jan Outrata
  • Vilem Vychodil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4923)


The paper presents additional results on factorization by similarity of fuzzy concept lattices. A fuzzy concept lattice is a hierarchically ordered collection of clusters extracted from tabular data. The basic idea of factorization by similarity is to have, instead of a possibly large original fuzzy concept lattice, its factor lattice. The factor lattice contains less clusters than the original concept lattice but, at the same time, represents a reasonable approximation of the original concept lattice and provides us with a granular view on the original concept lattice. The factor lattice results by factorization of the original fuzzy concept lattice by a similarity relation. The similarity relation is specified by a user by means of a single parameter, called a similarity threshold. Smaller similarity thresholds lead to smaller factor lattices, i.e. to more comprehensible but less accurate approximations of the original concept lattice. Therefore, factorization by similarity provides a trade-off between comprehensibility and precision. We first recall the notion of factorization. Second, we present a way to compute the factor lattice of a fuzzy concept lattice directly from input data, i.e. without the need to compute the possibly large original concept lattice.


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  1. 1.
    Belohlavek, R.: Fuzzy Galois connections. Math. Logic Quarterly 45(4), 497–504 (1999)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Belohlavek, R.: Similarity relations in concept lattices. J. Logic and Computation 10(6), 823–845 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Belohlavek, R.: Algorithms for fuzzy concept lattices. In: Proc. RASC 2002, Nottingham, UK, December 12–13, 2002, pp. 200–205 (2002)Google Scholar
  4. 4.
    Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer, Academic/Plenum Publishers, New York (2002)zbMATHGoogle Scholar
  5. 5.
    Belohlavek, R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Logic 128, 277–298 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Belohlavek, R., Dvořák, J., Outrata, J.: Fast factorization by similarity in formal concept analysis of data with fuzzy attributes. Journal of Computer and System Sciences 73(6), 1012–1022 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Belohlavek, R., Outrata, J., Vychodil, V.: On factorization by similarity of fuzzy concepts with hedges. Int. J. of Foundations of Computer Science (to appear)Google Scholar
  8. 8.
    Belohlavek, R., Vychodil, V.: Reducing the size of fuzzy concept lattices by hedges. In: FUZZ-IEEE 2005, The IEEE International Conference on Fuzzy Systems, Reno (Nevada, USA), May 22–25, 2005, pp. 663–668 (2005)Google Scholar
  9. 9.
    Belohlavek, R., Vychodil, V.: What is a fuzzy concept lattice? In: Proc. CLA 2005, 3rd Int. Conference on Concept Lattices and Their Applications, Olomouc, Czech Republic, September 7–9, 2005, pp. 34–45.,
  10. 10.
    Ben Yahia, S., Jaoua, A.: Discovering knowledge from fuzzy concept lattice. In: Kandel, A., Last, M., Bunke, H. (eds.) Data Mining and Computational Intelligence, pp. 167–190. Physica-Verlag, Heidelberg New York (2001)Google Scholar
  11. 11.
    Carpineto, C., Romano, G.: Concept Data Analysis. Theory and Applications. J. Wiley, Chichester (2004)zbMATHGoogle Scholar
  12. 12.
    Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer, Berlin (1999)zbMATHGoogle Scholar
  13. 13.
    Goguen, J.A.: The logic of inexact concepts. Synthese 18(9), 325–373 (1968)Google Scholar
  14. 14.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)zbMATHGoogle Scholar
  15. 15.
    Hájek, P.: On very true. Fuzzy Sets and Systems 124, 329–333 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice-Hall, Englewood Cliffs (1995)zbMATHGoogle Scholar
  17. 17.
    Krajči, S.: Cluster based efficient generation of fuzzy concepts. Neural Network World 5, 521–530 (2003)Google Scholar
  18. 18.
    Pollandt, S.: Fuzzy Begriffe. Springer, Berlin/Heidelberg (1997)zbMATHGoogle Scholar
  19. 19.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, Reidel, Dordrecht, Boston, pp. 445–470 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Radim Belohlavek
    • 1
    • 2
  • Jan Outrata
    • 2
  • Vilem Vychodil
    • 1
    • 2
  1. 1.Dept. Systems Science and Industrial Engineering T. J. Watson School of Engineering and Applied ScienceBinghamton University–SUNYBinghamtonUSA
  2. 2.Dept. Computer SciencePalacky UniversityOlomoucCzech Republic

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