Abstract
In order to modeling the interactions among attributes for classification, the non-additive measures are studied in this chapter. The non-additive measures provide very important information regarding the interactions among attributes and potentially useful for data mining. The concept of non-additive measures (also referred to as fuzzy measure theory) was initiated in the 1950s and have been well developed since 1970s. Nonadditive measures have been successfully used as a data aggregation tool for many applications such as information fusion, multiple regressions and classifications. The nonlinear integrals are the aggregation tools for the non-additive measures. The Choquet integral, a nonlinear integral, is utilized to aggregate the feature attributes with respect to the non-additive measure. The non-additive MCLP classification models are constructed in this chapter, and because the using of non-additive measure increases the computational cost, two major solutions to reduce the number of non-additive measures are given: hierarchical Choquet integral and the K-additive measure.
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References
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)
Denneberg, D.: Fuzzy Measure and Integral. Kluwer Academic, Dordrecht (1994)
Grabisch, M., Sugeno, M.: Multi-attribute classification using fuzzy integral. In: Proceedings of the FUZZ-IEEE’92, pp. 47–54 (1992)
Grabisch, M., Nicolas, J.M.: Classification by fuzzy integral: performance and tests. Fuzzy Sets Syst. 65, 255–271 (1994)
Grabisch, M.: A new algorithm for identifying fuzzy measures and its application to pattern recognition. In: Proceedings of 1995 IEEE International Conference on Fuzzy Systems (1995)
Grabisch, M.: The interaction and Möbius representation of fuzzy measures on finite spaces, k-additive measures: a survey. In: Fuzzy Measures and Integrals—Theory and Applications, pp. 70–93 (2000)
Han, J., Kamber, M.: Data Mining Concepts and Techniques. Morgan Kaufmann, San Mateo (2002)
Jgap: a genetic algorithms and genetic programming component by java (2008)
Keerthi, S., Shevade, S., Bhattacharyya, C., Murthy, K.: Improvements to Platt’s SMO algorithm for SVM classifier design. Technical report, Dept. of CSA, Banglore, India (1999)
Kou, G.: Multi-class multi-criteria mathematical programming and its applications in large scale data mining problems. PhD thesis, University of Nebraska Omaha (2006)
Liu, M., Wang, Z.: Classification using generalized Choquet integral projections. In: Proc. IFSA, pp. 421–426 (2005)
Mikenina, L., Zimmermann, H.J.: Improved feature selection and classification by the 2-additive fuzzy measure. Fuzzy Sets Syst. 107(2), 197–218 (1999)
Miranda, P., Grabisch, M.: Optimization issues for fuzzy measures. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 7, 545–560 (1999)
Murofushi, T., Sugeno, M., Fujimoto, K.: Separated hierarchical decomposition of the Choquet integral. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 5(5), 563–585 (1997)
Platt, J.: Fast training of support vector machines using sequential minimal optimization. Technical report, Microsoft Research (1998)
Sugeno, M., Fujimoto, K., Murofushi, T.: Hierarchical decomposition theorems for Choquet integral models. In: Proceedings of 1995 IEEE International Conference on Fuzzy Systems (1995)
Wang, Z., Guo, H.: A new genetic algorithm for nonlinear multiregressions based on generalized Choquet integrals. In: Proc. of FUZZ/IEEE, pp. 819–821 (2003)
Wang, Z., Klir, G.: Fuzzy Measure Theory. Plenum, New York (1992)
Wang, Z., Leung, K.-S., Klir, G.J.: Applying fuzzy measures and nonlinear integrals in data mining. Fuzzy Sets Syst. 156, 371–380 (2005)
Xu, K., Wang, Z., Heng, P., Leung, K.: Classification by nonlinear integral projections. IEEE Trans. Fuzzy Syst. 11, 187–201 (2003)
Yan, N., Wang, Z., Shi, Y., Chen, Z.: Nonlinear classification by linear programming with signed fuzzy measures. In: Proceedings FUZZ/IEEE, vol. 11, pp. 187–201 (2006)
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Shi, Y., Tian, Y., Kou, G., Peng, Y., Li, J. (2011). Non-additive MCLP. In: Optimization Based Data Mining: Theory and Applications. Advanced Information and Knowledge Processing. Springer, London. https://doi.org/10.1007/978-0-85729-504-0_10
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DOI: https://doi.org/10.1007/978-0-85729-504-0_10
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