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Information Fusion in Data Mining pp 9–26Cite as

On some aggregation operators for numerical information

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 123))

Abstract

In this chapter we review some of the aggregation operators that are appropriate for fusing numerical information. We focus on the ones that belong to Choquet and Sugeno integral families. This is, the ones that these integrals generalize. In particular, the review includes, among others, the following operators: arithmetic mean, weighted mean, OWA and WOWA operators, weighted minimum and weighted maximum, Sugeno and Choquet integral.

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References

  1. Aczél, J., (1984), On weighted synthesis of judgements, Aequationes Math., 27 288–307.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aczél, J., (1987), A short course on functional equations, D. Reidel Publishing Company.

    Book  MATH  Google Scholar 

  3. Aczél, J., Alsina, C., (1987), Characterization of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements, Methods of Operations Research, 48.

    Google Scholar 

  4. Beliakov, G., (2001), Shape preserving splines in constructing WOWA operators, Fuzzy Sets and Systems, 113 389–396.

    Google Scholar 

  5. Calvo, T., Mayor, G., Mesiar, R., (Eds.), (2002), Aggregation operators: New Trends and Applications, Physica-Verlag.

    MATH  Google Scholar 

  6. Calvo, T., Kolesárová, A., Komornfková, M., Mesiar, R., (2002), Aggregation Operators: Properties, Classes and Construction Methods, in T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation operators: New Trends and Applications, Physica-Verlag, 3–123.

    Chapter  Google Scholar 

  7. Carbonell, M., Mas, M., Mayor, G., (1997), On a class of Monotonic Extended OWA Operators, Proc. of the Sixth IEEE International Conference on Fuzzy Systems (IEEE-FUZZ′97), 1695–1699, Barcelona, Catalonia, Spain.

    Google Scholar 

  8. Choquet, G., (1953), Theory of Capacities, Ann. Inst. Fourier 5 131–296.

    Article  MathSciNet  Google Scholar 

  9. Dubois, D., Prade, H., (1986), Weighted Minimum and Maximum Operations in Fuzzy Set Theory, Information Sci. 39 205–210.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dubois, D., Prade, H., Testemale, C., (1988), Weighted fuzzy pattern-matching, Fuzzy Sets and Systems, 28 313–331.

    Article  MathSciNet  MATH  Google Scholar 

  11. Filev, D. P., Yager, R. R., (1998), On the issue of obtaining OWA operator weights, Fuzzy Sets and Systems, 94 157–169.

    Article  MathSciNet  Google Scholar 

  12. Fodor, J., Marichal, J. L., Roubens, M., (1995), Characterization of the Ordered Weighted Averaging Operators, IEEE Trans. on Fuzzy Systems, 3:2 236–240.

    Article  Google Scholar 

  13. Fodor, J., Roubens, M., (1994), Fuzzy preference modelling and multicriteria decision support, Kluwer Academic Publishers.

    Book  MATH  Google Scholar 

  14. Grabisch, M., (1995), Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69 279–298.

    Article  MathSciNet  MATH  Google Scholar 

  15. Grabisch, M., (1996), K-order additive fuzzy measures, Proc. 6th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), 1345–1350, Granada, Spain.

    Google Scholar 

  16. Grabisch, M., (1998), Fuzzy Integral as a Flexible and Interpretable Tool of Aggregation, in B. Bouchon-Meunier, (Ed.), Aggregation and Fusion of Imperfect Information, Physica-Verlag, 51–72.

    Chapter  Google Scholar 

  17. Grabisch, M., (2000), The Interaction and Möbius Representation of Fuzzy Measures on Finite Spaces, k-Additive Measures: A Survey, in M. Grabisch, T. Murofushi, M. Sugeno, (Eds.), Fuzzy Measures and Integrals: Theory and Applications, Physica-Verlag, 70–93.

    Google Scholar 

  18. Grabisch, M., (2002), Aggregation Based on Integrals: Recent Results and Trends, in T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation operators: New Trends and Applications, Physica-Verlag, 107–123.

    Chapter  Google Scholar 

  19. Grabisch, M., (2003), Modelling data by the Choquet integral, Chapter in this book.

    Google Scholar 

  20. Grabisch, M., Nguyen, H. T., Walker, E.A., (1995), Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference, Kluwer Academic Publishers, Dordreht, The Netherlands.

    Book  Google Scholar 

  21. Grabisch, M., Murofushi, T., Sugeno, M., (Eds.), (2000), Fuzzy Measures and Integrals: Theory and Applications, Physica-Verlag.

    MATH  Google Scholar 

  22. Honda, A., Nakano, T., Okazaki, Y., (2002), Subjective evaluation based on distorted probability, Proc. of the SCIS & ISIS Conference (CD-ROM), Tsukuba, Japan.

    Google Scholar 

  23. Honda, A., Nakano, T., Okazaki, Y., (2002), Distortion of Fuzzy Measures, Proc. of the SCIS & ISIS Conference (CD-ROM), Tsukuba, Japan.

    Google Scholar 

  24. Imai, H., Miyamori, M., Miyakosi, M., Sato, Y., (2000), An algorithm Based on Alternative Projections for a Fuzzy Measures Identification Problem, Proc. of the Iizuka Conference, lizuka, Japan (CD-Rom).

    Google Scholar 

  25. Imai, H., Asano, D., Sato, Y., (2003), An Algorithm Based on Alternative Projections for a Fuzzy Measure Identification Problem, Chapter in this book.

    Google Scholar 

  26. Kandel, A., Byatt, W. J., (1978), Fuzzy sets, fuzzy algebra, and fuzzy statistics, Proc. of the IEEE 66 1619–1639.

    Article  Google Scholar 

  27. Klir, G. J., Yuan, B., (1995), Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, U.K..

    MATH  Google Scholar 

  28. Marichal, J.-L., (1999), Aggregation operators for multicriteria decision aid, PhD Dissertation, Faculte des Sciences, Universite de Liege, Liège, Belgium.

    Google Scholar 

  29. Marichal, J.-L., Roubens, M., (2000), Determination of weights of interacting criteria from a reference set, European Journal of Operational Research 124:3 641–650.

    Article  MathSciNet  MATH  Google Scholar 

  30. Murofushi, T., Sugeno, M., (1989), An interpretation of fuzzy measure and Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems, 29 201–227

    Article  MathSciNet  MATH  Google Scholar 

  31. Murofushi, T., Sugeno, M., (1991), Fuzzy t-conorm integral with respect to fuzzy measures: generalization of Sugeno integral and Choquet integral, Fuzzy Sets and Systems, 42:1 57–71.

    Article  MathSciNet  MATH  Google Scholar 

  32. Murofushi, T., Sugeno, M, (1993), Some quantities represented by the Choquet integral, Fuzzy Sets and Systems, 56 229–235.

    Article  MathSciNet  MATH  Google Scholar 

  33. Murofushi, T., Sugeno, M., (2000), Fuzzy Measures and Fuzzy Integrals, in M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Fuzzy Measures and Integrals: Theory and Applications, Physica-Verlag, 3–41.

    Google Scholar 

  34. Narukawa, Y., Murofushi, T., (2003), Choquet integral and Sugeno integral as aggregation functions, Chapter of this book.

    Google Scholar 

  35. Nettleton, D., Torra, V., (2001), A comparision of active set method and genetic algorithm approaches for learning weighting vectors in some aggregation operators, Int. J. of Intel. Syst., 16 1069–1083.

    Article  MATH  Google Scholar 

  36. O’Hagan, M., (1988), Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic, Proceedings of the 22nd Annual IEEE Asilomar Conference on Signals, Systems and Computers, 681–689, Pacific Grove, CA, USA.

    Google Scholar 

  37. Saaty, T. L., (1980), The Analytic Hierarchy Process, McGraw-Hill.

    MATH  Google Scholar 

  38. Sugeno, M., (1974), Theory of fuzzy integrals and its applications, Ph. D. dissertation, Tokyo Institute of Technology.

    Google Scholar 

  39. Sugeno, M., Murofushi, T., (1993), Fuzzy Measure, Tokyo, Nikkan Kogyo Shinbunsha (in Japanese).

    Google Scholar 

  40. Tanaka, A., Murofushi, T., (1989), A learning model using fuzzy measure and the Choquet integral, Proc. of the 5th Fuzzy System Symposium, 213–217, Kobe, Japan (in Japanese).

    Google Scholar 

  41. Torra, V., (1996), Weighted OWA operators for synthesis of information, Proc. of Fifth IEEE Int. Conference on Fuzzy Systems (IEEE-FUZZ′96) (ISBN 0–7803-3645–3), 966–971, New Orleans, USA.

    Google Scholar 

  42. Torra, V., (1997), The Weighted OWA operator, Int. J. of Intel. Syst., 12 153–166.

    Article  MATH  Google Scholar 

  43. Torra, V., (1998), On some relationships between the WOWA operator and the Choquet integral, Proc. of the Seventh Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU) (ISBN 2–84254-013–1), 818–824, Paris, France.

    Google Scholar 

  44. Torra, V., (1998), On considering constraints of different importance in fuzzy constraint satisfaction problems, Int. J. of Uncertain Fuzziness and Knowledge-Bases 6:5 489–501.

    Article  MATH  Google Scholar 

  45. Torra, V., (2000), The WOWA operator and the interpolation function W*: Chen and Otto’s interpolation method revisited, Fuzzy Sets and Systems, 113:3 389–396.

    Article  MathSciNet  Google Scholar 

  46. Torra, V., (1998), On the integration of numerical information: from the arithmetic mean to fuzzy integrals, Report de Recerca, DEI-RR-98–010, Tarragona, Universitat Rovira i Virgili.

    Google Scholar 

  47. Torra, V., (1999), On hierarchically S-decomposable fuzzy measures, Int. J. of Intel. Syst., 14:9 923–934.

    Article  MATH  Google Scholar 

  48. Torra, T., (1999), On the learning of weights in some aggregation operators: The weighted mean and the OWA operators. Mathware and Soft Computing, 6 249–265.

    MathSciNet  Google Scholar 

  49. Torra, V., (2000), Learning Weights for Weighted OWA Operators, Proc. of the IEEE Int. Conference on Industrial Electronics, Control and Instrumentation (IECON 2000), CD-ROM, Nagoya, Japan.

    Google Scholar 

  50. Torra, V., (2001), Authors’s reply (to an article by G. Beliakov), Fuzzy Sets and Systems, 121 551.

    Article  MathSciNet  MATH  Google Scholar 

  51. Torra, V., (2002), http://www.iiia.csic.es/~vtorra/wowa/wowa.html

  52. Torra, V., (2002), Learning weights for the quasi-weighted means, IEEE Trans. on Fuzzy Systems, 10:5 653–666.

    Article  MathSciNet  Google Scholar 

  53. Wang, Z., Klir, G., (1992), Fuzzy measure theory, Plenum Press.

    Book  MATH  Google Scholar 

  54. Yager, R.R., (1988), On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans. on SMC, 18 183–190.

    Article  MathSciNet  MATH  Google Scholar 

  55. Yager, R.R., (1993), Families of OWA operators, Fuzzy Sets and Systems, 59 125–148.

    Article  MathSciNet  MATH  Google Scholar 

  56. Yager, R. R., (1996), Quantifier Guided Aggregation Using OWA operators, Int. J. of Int. Syst., 11 49–73.

    Article  Google Scholar 

  57. Yager, R. R., Kacprzyk, J., (1997), The Ordered Weighting Averaging Operators, Kluwer Academic Publishers.

    Book  Google Scholar 

  58. Yoneda, M., Fukami, S., Grabisch, M., (1994), Human factor and fuzzy science, in K. Asai (Ed.), Fuzzy science, Kaibundo, 93–122 (in Japanese).

    Google Scholar 

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Authors and Affiliations

  1. Institut d’Investigació en Intel•ligència Artificial - CSIC, Campus UAB s/n, 08193, Bellaterra, Catalonia, Spain

    Vicenç Torra

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  1. Vicenç Torra
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  1. Institut d’Investigació en Intel·ligència Artificial CSIC - Spanish Scientific Research Council, Campus Universitat Autònoma de Barcelona, 08193, Bellaterra, Catalonia, Spain

    Vicenç Torra

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Torra, V. (2003). On some aggregation operators for numerical information. 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_2

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  • DOI: https://doi.org/10.1007/978-3-540-36519-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-05628-4

  • Online ISBN: 978-3-540-36519-8

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