Factorizing Three-Way Ordinal Data Using Triadic Formal Concepts

  • Radim Belohlavek
  • Petr Osička
  • Vilem Vychodil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7022)

Abstract

The paper presents a new approach to factor analysis of three-way ordinal data, i.e. data described by a 3-dimensional matrix I with values in an ordered scale. The matrix describes a relationship between objects, attributes, and conditions. The problem consists in finding factors for I, i.e. finding a decomposition of I into three matrices, an object-factor matrix A, an attribute-factor matrix B, and a condition-factor matrix C, with the number of factors as small as possible. The difference from the decomposition-based methods of analysis of three-way data consists in the composition operator and the constraint on A, B, and C to be matrices with values in an ordered scale. We prove that optimal decompositions are achieved by using triadic concepts of I, developed within formal concept analysis, and provide results on natural transformations between the space of attributes and conditions and the space of factors. We present an illustrative example demonstrating the usefulness of finding factors and a greedy algorithm for computing decompositions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belohlavek, R.: Optimal decomposition of matrices with entries from residuated lattices. J. Logic and Computation (to appear, preliminary version appeared in Proc. IEEE Intelligent Systems, pp. 15-2–15-7 (2008))Google Scholar
  2. 2.
    Belohlavek, R., Glodeanu, C.V., Vychodil, V.: Optimal factorization of three-way binary data using triadic concepts. (submitted, preliminary version appeared in Proc. IEEE GrC 2010, pp. 61–66 (2010))Google Scholar
  3. 3.
    Belohlavek, R., Osicka, P.: Triadic concept analysis of data with fuzzy attributes. In: Proc. 2010 IEEE International Conference on Granular Computing, San Jose, California, August 14–16, pp. 661–665 (2010)Google Scholar
  4. 4.
    Belohlavek, R., Vychodil, V.: Discovery of optimal factors in binary data via a novel method of matrix decomposition. J. Computer and System Sci. 76(1), 3–20 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Belohlavek, R., Vychodil, V.: Factor analysis of incidence data via novel decomposition of matrices. In: Ferré, S., Rudolph, S. (eds.) ICFCA 2009. LNCS(LNAI), vol. 5548, pp. 83–97. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Cichocki, A., et al.: Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. J. Wiley, Chichester (2009)CrossRefGoogle Scholar
  7. 7.
    Cormen, T.H., et al.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  8. 8.
    Ganter, B., Wille, R.: Formal Concept Analysis. Mathematical Foundations. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)CrossRefMATHGoogle Scholar
  10. 10.
    Jäschke, R., Hotho, A., Schmitz, C., Ganter, B., Stumme, G.: TRIAS – An Algorithm for Mining Iceberg Tri-Lattices. In: ICDM 2006, pp. 907–911 (2006)Google Scholar
  11. 11.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice-Hall, Englewood Cliffs (1995)MATHGoogle Scholar
  12. 12.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Review 51(3), 455–500 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kroonenberg, P.M.: Applied Multiway Data Analysis. J. Wiley, Chichester (2008)CrossRefMATHGoogle Scholar
  14. 14.
    Kuznetsov, S., Obiedkov, S.: Comparing performance of algorithms for generating concept lattices. J. Experimental and Theoretical Articial Intelligence 14(2–3), 189–216 (2002)CrossRefMATHGoogle Scholar
  15. 15.
    Lehmann, F., Wille, R.: A triadic approach to formal concept analysis. In: Ellis, G., Rich, W., Levinson, R., Sowa, J.F. (eds.) ICCS 1995. LNCS, vol. 954, pp. 32–34. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  16. 16.
    Mickey, M.R., Mundle, P., Engelman, L.: Boolean factor analysis. In: Dixon, W.J. (ed.) BMDP Statistical Software Manual, vol. 2, pp. 849–860. University of California Press, Berkeley (1990)Google Scholar
  17. 17.
    Miettinen, P., Mielikäinen, T., Gionis, A., Das, G., Mannila, H.: The Discrete Basis Problem. In: Fürnkranz, J., Scheffer, T., Spiliopoulou, M. (eds.) PKDD 2006. LNCS (LNAI), vol. 4213, pp. 335–346. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Nau, D.S., Markowsky, G., Woodbury, M.A., Amos, D.B.: A Mathematical Analysis of Human Leukocyte Antigen Serology. Math. Biosciences 40, 243–270 (1978)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Outrata, J.: Preprocessing input data for machine learning by FCA. In: Kryszkiewicz, M., Obiedkov, S. (eds.) Proc. CLA 2010, vol. 672, pp. 187–198. University of Sevilla, CEUR WS (2010)Google Scholar
  20. 20.
    Outrata, J.: Boolean factor analysis for data preprocessing in machine learning. In: Draghici, S., et al. (eds.) Proc. ICMLA 2010, Intern. Conf. on Machine Learning and Applications, pp. 899–902. IEEE, Washington, DC (2010)CrossRefGoogle Scholar
  21. 21.
    Smilde, A., Bro, R., Geladi, P.: Multi-way Analysis: Applications in the Chemical Sciences. J. Wiley, Chichester (2004)CrossRefGoogle Scholar
  22. 22.
    Stockmeyer, L.J.: The set basis problem is NP-complete. IBM Research Report RC5431, Yorktown Heights, NY (1975)Google Scholar
  23. 23.
    Tang, F., Tao, H.: Binary principal component analysis. In: Proc. British Machine Vision Conference 2006, pp. 377–386 (2006)Google Scholar
  24. 24.
    Tatti, N., Mielikäinen, T., Gionis, A., Mannila, H.: What is the dimension of your binary data? In: ICDM 2006, pp. 603–612 (2006)Google Scholar
  25. 25.
    Wille, R.: The basic theorem of triadic concept analysis. Order 12, 149–158 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Radim Belohlavek
    • 1
  • Petr Osička
    • 1
  • Vilem Vychodil
    • 1
  1. 1.Department of Computer Science, Data Analysis and Modeling LabPalacky UniversityOlomoucCzech Republic

Personalised recommendations