Abstract
The chapter makes a survey of works dealing with the Choquet integral as a general non linear regression model. It is shown that its use is however limited to commensurate variables, as it is the case for example for multicriteria evaluation and multiattribute classification. A large part is devoted to the various methods of identifying parameters of the model, essentially quadratic programming and genetic algorithms. A new approach based on genetic algorithms is also described. Lastly, related works on classification and subjective evaluation are mentionned.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
B.D.O. Anderson and J.B. Moore. Optimal filtering. Prentice Hall, 1979.
A. Chateauneuf. Modeling attitudes towards uncertainty and risk through the use of Choquet integral. Annals of Operations Research, 52:3–20, 1994.
T.Y. Chen and J.C. Wang. Identification of A-measures using sampling design and genetic algorithms. Fuzzy Sets and Systems, 123:321–341, 2001.
G. Choquet. Theory of capacities. Annales de l’Institut Fourier, 5:131–295, 1953.
D. Denneberg. Non-Additive Measure and Integral. Kluwer Academic, 1994.
K. Fujimoto and T. Murofushi. Hierarchical decomposition of the Choquet integral. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 94–104. Physica Verlag, 2000.
M. Grabisch. A new algorithm for identifying fuzzy measures and its application to pattern recognition. In Int. Joint Conf. of the 4th IEEE Int. Conf. on Fuzzy Systems and the 2nd Int. Fuzzy Engineering Symposium, pages 145–150, Yokohama, Japan, March 1995.
M. Grabisch. The application of fuzzy integrals in multicriteria decision making. European J. of Operational Research, 89:445–456, 1996.
M. Grabisch. The representation of importance and interaction of features by fuzzy measures. Pattern Recognition Letters, 17:567–575, 1996.
M. Grabisch. Alternative representations of discrete fuzzy measures for decision making. Int. J. of Uncertainty, Fuzziness, and Knowledge Based Systems, 5:587–607, 1997.
M. Grabisch. K-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, 92:167–189, 1997.
M. Grabisch. Fuzzy integral for classification and feature extraction. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 415–434. Physica Verlag, 2000.
M. Grabisch. A graphical interpretation of the Choquet integral. IEEE Tr. on Fuzzy Systems, 8:627–631, 2000.
M. Grabisch, J.M. Baret, and M. Larnicol. Analysis of interaction between criteria by fuzzy measure and its application to cosmetics. In Int. Conf. on Methods and Applications of Multicriteria Decision Making, pages 22–25, Mons, Belgium, May 1997.
M. Grabisch, J. Duchêne, F. Lino, and P. Perny. Subjective evaluation of discomfort in sitting position. Fuzzy Optimization and Decision Making, 1:287–312, 2002.
M. Grabisch and Ch. Labreuche. To be symmetric or asymmetric? A dilemna in decision making. In J. Fodor, B. De Baets, and P. Perny, editors, Preferences and Decisions under Incomplete Knowledge, pages 179–194. Physica Verlag, 2000.
M. Grabisch, T. Murofushi, and M. Sugeno. Fuzzy Measures and Integrals. Theory and Applications (edited volume). Studies in Fuzziness. Physica Verlag, 2000.
M. Grabisch, H.T. Nguyen, and E.A. Walker. Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference. Kluwer Academic, 1995.
M. Grabisch and J.M. Nicolas. On the performance of classification techniques based on fuzzy integrals. In 5th Int. Fuzzy Systems Assoc. Congress, pages 163–166, Seoul, Korea, July 1993.
M. Grabisch and J.M. Nicolas. Classification by fuzzy integral — performance and tests. Fuzzy Sets & Systems, Special Issue on Pattern Recognition, 65:255–271, 1994.
M. Grabisch and M. Roubens. An axiomatic approach to the concept of interaction among players in cooperative games. Int. Journal of Game Theory, 28:547–565, 1999.
M. Grabisch and M. Sugeno. Fuzzy integral with respect to dual measures and its application to multi-attribute pattern recognition. In 6th Fuzzy Systems Symposium, pages 205–209, Tokyo, Japan, September 1990. in japanese.
M. Grabisch and M. Sugeno. Multi-attribute classification using fuzzy integral. In 1st IEEE Int. Conf. on Fuzzy Systems, pages 47–54, San Diego, CA, March 1992.
J.M. Keller, P.D. Gader, and A.K. Hocaoglu. Fuzzy integrals in image processing and recognition. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 435–466. Physica Verlag, 2000.
S.H. Kwon and M. Sugeno. A hierarchical subjective evaluation model using non-monotonic fuzzy measures and the Choquet integral. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals — Theory and Applications, pages 375–391. Physica Verlag, 2000.
P. Miranda and M. Grabisch. Optimization issues for fuzzy measures. Int. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems, 7(6):545–560, 1999.
T. Mori and T. Murofushi. An analysis of evaluation model using fuzzy measure and the Choquet integral. In 5th Fuzzy System Symposium, pages 207–212, Kobe, Japan, 1989. In Japanese.
T. Murofushi and S. Soneda. Techniques for reading fuzzy measures (III): interaction index. In 9th Fuzzy System Symposium, pages 693–696, Sapporo, Japan, May 1993. In Japanese.
D. Nettleton and V. Torra. A comparison of active set method and genetic algorithm approaches for learning weighting vectors in some aggregation operators. Int. J. of Intelligent Systems, 16(9):1069–1083, 2001.
T. Onisawa, M. Sugeno, Y. Nishiwaki, H. Kawai, and Y. Harima. Fuzzy measure analysis of public attitude towards the use of nuclear energy. Fuzzy Sets & Systems, 20:259–289, 1986.
E. Pap. Null-Additive Set Functions. Kluwer Academic, 1995.
D. Schmeidler. Subjective probability and expected utility without additivity. Econometrica, 57(3):571–587, 1989.
U. Schmidt. Axiomatic Utility Theory under Risk. Number 461 in Lectures Notes in Economics and Mathematical Systems. Springer Verlag, 1998.
L.S. Shapley. A value for n-person games. In H.W. Kuhn and A.W. Tucker, editors, Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, pages 307–317. Princeton University Press, 1953.
M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.
M. Sugeno and S.H. Kwon. A clusterwise regression-type model for subjective evaluation. J. of Japan Society for Fuzzy Theory and Systems, 7(2):291–310, 1995.
M. Sugeno and S.H. Kwon. A new approach to time series modeling with fuzzy measures and the Choquet integral. In Int. Joint Conf. of the 4th IEEE Int. Conf. on Fuzzy Systems and the 2nd Int. Fuzzy Engineering Symp., pages 799–804, Yokohama, Japan, March 1995.
H. Tahani and J.M. Keller. Information fusion in computer vision using the fuzzy integral. IEEE Tr. on Systems, Man, and Cybernetics, 20(3):733–741, 1990.
Z. Wang and G.J. Klir. Fuzzy measure theory. Plenum, 1992.
Z. Wang, K.S. Leung, and J. Wang. A genetic algorithm for determining nonadditive set functions in information fusion. Fuzzy Sets and Systems, 102:462–469, 1999.
Z. Wang, K.S. Leung, M.L. Wong, J. Fang, and K. Xu. Nonlinear nonnegative multiregressions based on Choquet integrals. Int. J. of Approximate Reasoning, 25:71–87, 2000.
K. Xu, Z. Wang, and K.S. Leung. Using a new type of nonlinear integral for multiregression: an application of evolutionary algorithms in data mining. In Proc. of IEEE Int. Conf. on Systems, Man and Cybernetics, pages 2326–2331, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Grabisch, M. (2003). Modelling data by the Choquet integral. In: Torra, V. (eds) Information Fusion in Data Mining. Studies in Fuzziness and Soft Computing, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36519-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-36519-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05628-4
Online ISBN: 978-3-540-36519-8
eBook Packages: Springer Book Archive