Abstract
In this contribution, we introduce special transformations of fusion functions closely related to ratio scale, difference scale and interval scale invariant fusion functions. In particular, we show that in the case of aggregation functions the obtained transforms need not be aggregation functions but they are always pre-aggregation functions. We also provide sufficient conditions for constructing aggregation functions which are ratio scale, difference scale or interval scale invariant. We illustrate the obtained results applying the introduced transformations to the basic fuzzy integrals. It is shown that only the Choquet integral is invariant with respect to all studied transformations.
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References
Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Directional monotonicity of fusion functions. Eur. J. Oper. Res. 244(1), 300–308 (2015)
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer, Berlin (2007)
Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press (2009)
Lucca, G., Sanz, J., Dimuro, G.P., Bedregal, B., Mesiar, R., Kolesárová, A., Bustince, H.: Pre-aggregation functions: construction and an application. IEEE Trans. Fuzzy Syst. 24, 260–272 (2016)
Wilkin, T., Beliakov, G.: Weakly monotone aggregation functions. Int. J. Intell. Syst. 30, 144–165 (2015)
Lázaro, J., Rückschlossová, T., Calvo, T.: Shift invariant binary aggregation operators. Fuzzy Sets Syst. 142(1), 51–62 (2004)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953)
Sugeno, M.: Theory of Fuzzy Integrals and Applications. Ph.D. Thesis, Tokyo Institute of Technology, Tokyo, 1974
Shilkret, N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)
Acknowledgements
The authors kindly acknowledge the support of the projects APVV-18-0052, APVV-17-0066 and the grant VEGA 1/0614/18.
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Kolesárová, A., Mesiar, R. (2022). Invariant Aggregation and Pre-aggregation Functions. In: Harmati, I.Á., Kóczy, L.T., Medina, J., Ramírez-Poussa, E. (eds) Computational Intelligence and Mathematics for Tackling Complex Problems 3. Studies in Computational Intelligence, vol 959. Springer, Cham. https://doi.org/10.1007/978-3-030-74970-5_3
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DOI: https://doi.org/10.1007/978-3-030-74970-5_3
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