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Efficient Learning of Classifiers Based on the 2-Additive Choquet Integral

  • Eyke Hüllermeier
  • Ali Fallah Tehrani
Part of the Studies in Computational Intelligence book series (SCI, volume 445)

Abstract

In a recent work, we proposed a generalization of logistic regression based on the Choquet integral. Our approach, referred to as choquistic regression, makes it possible to capture non-linear dependencies and interactions among predictor variables while preserving two important properties of logistic regression, namely the comprehensibility of the model and the possibility to ensure its monotonicity in individual predictors. Unsurprisingly, these benefits come at the expense of an increased computational complexity of the underlying maximum likelihood estimation. In this paper, we propose two approaches for reducing this complexity in the specific though practically relevant case of the 2-additive Choquet integral. Apart from theoretical results, we also present an experimental study in which we compare the two variants with the original implementation of choquistic regression.

Keywords

Logistic Regression Extreme Point Sequential Quadratic Programming Fuzzy Measure Original Implementation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Angilella, S., Greco, S., Matarazzo, B.: Non-additive robust ordinal regression with Choquet integral, bipolar and level dependent Choquet integrals. In: Proc. IFSA/EUSFLAT 2009, Lisbon, Portugal, pp. 1194–1199 (2009)Google Scholar
  2. 2.
    Beliakov, G., James, S.: Citation-based journal ranks: the use of fuzzy measures. Fuzzy Sets and Systems 167(1), 101–119 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ben-David, A.: Monotonicity maintenance in information-theoretic machine learning algorithms. Machine Learning 19, 29–43 (1995)Google Scholar
  4. 4.
    Choquet, G.: Theory of capacities. Annales de l’institut Fourier 5, 131–295 (1954)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Daniels, H., Kamp, B.: Applications of MLP networks to bond rating and house pricing. Neural Computation and Applications 8, 226–234 (1999)CrossRefGoogle Scholar
  6. 6.
    David, A.B.: (2010), http://mldata.org/repository/data/viewslug/datasets-arie_ben_david-swd/ (last accessed on June 21, 2012)
  7. 7.
    Tehrani, A.F., Cheng, W., Dembczy, K., Hüllermeier, E.: Learning Monotone Nonlinear Models Using the Choquet Integral. In: Gunopulos, D., Hofmann, T., Malerba, D., Vazirgiannis, M. (eds.) ECML PKDD 2011. LNCS, vol. 6913, pp. 414–429. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Fallah Tehrani, A., Cheng, W., Hüllermeier, E.: Choquistic regression: Generalizing logistic regression using the Choquet integral. In: 7th Int. Conf. of the European Society for Fuzzy Logic and Technology, EUSFLAT 2011, Aix-les-Bains, France, pp. 868–875 (2011)Google Scholar
  9. 9.
    Fallah Tehrani, A., Cheng, W., Hüllermeier, E.: Preference learning using the Choquet integral: The case of multipartite ranking. IEEE Transactions on Fuzzy Systems (forthcoming, 2012)Google Scholar
  10. 10.
    Feelders, A.: Monotone relabeling in ordinal classification. In: Proc. of the 10th IEEE International Conference on Data Mining, pp. 803–808. IEEE Press, Piscataway (2010)CrossRefGoogle Scholar
  11. 11.
    Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems 69(3), 279–298 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Grabisch, M.: Modelling data by the Choquet integral. In: Torra, V. (ed.) Information Fusion in Data Mining, pp. 135–148. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    Grabisch, M., Nicolas, J.M.: Classification by fuzzy integral: performance and tests. Fuzzy Sets and Systems 65(2-3), 255–271 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measures and Integrals: Theory and Applications. Physica-Verlag, Heidelberg (2000)zbMATHGoogle Scholar
  15. 15.
    Miranda, P., Grabisch, M.: On vertices of the k-additive monotone core. In: Proc. IFSA/EUSFLAT 2009, Lisbon, Portugal, pp. 76–81 (2009)Google Scholar
  16. 16.
    Miranda, P., Grabisch, M., Gil, P.: Axiomatic structure of k-additive capacities. Mathematical Social Sciences 49, 153–178 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Miranda, P., Combarro, E.F., Gil, P.: Extreme points of some families of non-additive measures. European J. of Operational Research 174, 1865–1884 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Modave, F., Grabisch, M.: Preference representation by a Choquet integral: commensurability hypothesis. In: Proc. of the 7th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 164–171 (1998)Google Scholar
  19. 19.
    Potharst, R., Feelders, A.: Classification trees for problems with monotonicity constraints. ACM SIGKDD Explorations Newsletter 4(1), 1–10 (2002)CrossRefGoogle Scholar
  20. 20.
    Sugeno, M.: Theory of fuzzy integrals and its application. PhD thesis, Tokyo Institute of Technology, Tokyo, Japan (1974)Google Scholar
  21. 21.
    Torra, V.: Learning aggregation operators for preference modeling. In: Fürnkranz, J., Hüllermeier, E. (eds.) Preference Learning, pp. 317–333. Springer, Heidelberg (2011)Google Scholar
  22. 22.
    Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. Springer, Heidelberg (2007)Google Scholar
  23. 23.
    Vitali, G.: Sulla definizione di integrale delle funzioni di una variabile. Annali di Matematica Pura ed Applicata 2(1), 111–121 (1925)MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of MarburgMarburgGermany

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