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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg (2007)
Beliakov, G., Bustince, H., Calvo, T.: A Practical Guide to Averaging Functions. Springer, Heidelberg (2016)
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)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)
Grabisch, M.: \(k\)-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92, 167–189 (1997)
Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)
Kolesárová, A., Stupňanová, A., Beganová, J.: Aggregation-based extensions of fuzzy measures. Fuzzy Sets Syst. 194, 1–14 (2012)
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)
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)
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)
Valášková, L.: A note to the 2-order additivity. In: Proceedings of MAGIA, Kočovce, pp. 53–55 (2001)
Valášková, L.: Non-additive measures and integrals. Ph.D. thesis, STU Bratislava, (2007)
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-45656-0_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45655-3
Online ISBN: 978-3-319-45656-0
eBook Packages: Computer ScienceComputer Science (R0)