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On k–additive Aggregation Functions

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Modeling Decisions for Artificial Intelligence (MDAI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9880))

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Abstract

Inspired by the Grabisch idea of k–additive measures, we introduce and study k–additive aggregation functions. The Owen multilinear extension of a k–additive capacity is shown to be a particular k–additive aggregation function. We clarify the relation between k–additive aggregation functions and polynomials of a degree not exceeding k. We also describe \(n^2 + 2n\) basic 2–additive n–ary aggregation functions whose convex closure forms the class of all 2–additive n–ary aggregation functions.

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References

  1. Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  2. Beliakov, G., Bustince, H., Calvo, T.: A Practical Guide to Averaging Functions. Springer, Heidelberg (2016)

    Book  Google Scholar 

  3. Chateauneuf, A., Jaffray, J.Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Math. Soc. Sci. 17, 263–283 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grabisch, M.: \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92, 167–189 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  7. Kolesárová, A., Stupňanová, A., Beganová, J.: Aggregation-based extensions of fuzzy measures. Fuzzy Sets Syst. 194, 1–14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lovász, L.: Submodular function and convexity. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming: The state of the art, pp. 235–257. Springer, Berlin (1983)

    Chapter  Google Scholar 

  9. Marichal, J.-L.: Aggregation of interacting criteria by means of the discrete Choquet integral. In: Calvo, T., Mayor, G., Mesiar, R. (eds.) Aggregation Operators. New Trends and Applications, pp. 224–24. Physica-Verlag, Heidelberg (2002)

    Google Scholar 

  10. Owen, G.: Multilinear extensions of games. In: Shapley, S., Roth, A.E. (eds.) The Shapley value. Essays in Honour of Lloyd, pp. 139–151. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  11. Valášková, L.: A note to the 2-order additivity. In: Proceedings of MAGIA, Kočovce, pp. 53–55 (2001)

    Google Scholar 

  12. Valášková, L.: Non-additive measures and integrals. Ph.D. thesis, STU Bratislava, (2007)

    Google Scholar 

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Acknowledgement

A. Kolesárová and R. Mesiar kindly acknowledge the support of the project of Science and Technology Assistance Agency under the contract No. APVV–14–0013. J. Li acknowledges the support of the National Natural Science Foundation of China (Grants No. 11371332 and No. 11571106).

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Authors and Affiliations

  1. Faculty of Chemical and Food Technology, Slovak University of Technology, Radlinského 9, 812 37, Bratislava, Slovakia

    Anna Kolesárová

  2. School of Science, Communication University of China, 100024, Beijing, People’s Republic of China

    Jun Li

  3. Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05, Bratislava, Slovakia

    Radko Mesiar

Authors
  1. Anna Kolesárová
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  2. Jun Li
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  3. Radko Mesiar
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Corresponding author

Correspondence to Radko Mesiar .

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Editors and Affiliations

  1. University of Kövde , Skövde, Sweden

    Vicenç Torra

  2. Toho Gakuen , Kunitachi, Tokyo, Japan

    Yasuo Narukawa

  3. Enginyeria de la Informacio I de les Com, Univ Autonoma de Barcelona, Bellaterra, Spain

    Guillermo Navarro-Arribas

  4. Universitat d'Andorra , Sant Julià de Lòria, Andorra

    Cristina Yañez

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Kolesárová, A., Li, J., Mesiar, R. (2016). On k–additive Aggregation Functions. In: Torra, V., Narukawa, Y., Navarro-Arribas, G., Yañez, C. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2016. Lecture Notes in Computer Science(), vol 9880. Springer, Cham. https://doi.org/10.1007/978-3-319-45656-0_4

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  • DOI: https://doi.org/10.1007/978-3-319-45656-0_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45655-3

  • Online ISBN: 978-3-319-45656-0

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