Abstract
We bring an overview of fuzzy integrals, including historical remarks. Choquet integral can be traced back to 1925. Sugeno integral has a predecessor in Shilkret integral from 1971. Some other fuzzy integrals and the corresponding discrete integrals are also given. An application of Choquet integral to additive impreciseness measuring of fuzzy quantities with interesting consequences for fuzzy measures is presented. Finally, recent development and streaming of fuzzy integrals theory are mentioned.
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
Preview
Unable to display preview. Download preview PDF.
References
Benvenuti, P., Mesiar, R.: A note on Sugeno and Choquet integrals. In: Proc. IPMU (2000) , Madrid, pp. 582–585 (2000)
Benvenuti, P., Mesiar, R., Vivona, D.: Monotone Set Functions-Based Integrals. In:[26] (2002)
Benvenuti, P., Mesiar, R.: Pseudo-arithmetical operations as a basis for the general measure and integration theory. Information Sciences 160, 1–11 (2004)
Benvenuti, P., Mesiar, R.: On Tarski’s contribution to the additive measure theory and its consequences. Annals of Pure and Applied Logic 126, 281–286 (2004)
Bouchon-Meunier, B., Mesiar, R., Ralescu, D.: Linear Non-additive Set- Functions. Int. J. of General Systems 33(1), 89–98 (2004)
Calvo, T., Mayor, G., Mesiar, R. (eds.): Aggregation operators. New Trends and Applications. Physica, Heidelberg (2002)
Calvo, T., Mesiarová, A., Valášková, L.: Construction of Aggregation Operators: New Composition Method. Kybernetika 39, 643–650 (2003)
de Campos, L.M., Bolaňos, M.J.: Characterization and comparison of Sugeno and Choquet integrals. Fuzzy Sets and Systems 52, 61–67 (1993)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5(8) , 131–295 (1953-1954)
Denneberg, D.: Non-additive Measure and Integral. Kluwer Acad. Publ., Dordrecht (1994)
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)
Dubois, D., Prade, H.: Representation and combination of uncertainty with belief functions and possibility measures. Computational Intelligence 4, 244–264 (1988)
Grabisch, M., Murofushi, T., Sugeno, M.: Fuzzy Measures and Integrals. Physica, Heidelberg (2000)
Grabisch, M.: The symmetric Sugeno integral. Fuzzy Sets and Systems 139, 473–490 (2003)
Grabisch, M., Labreuche, C.: Bi-capacities for decision making on bipolar scales. In: Proc. EUROFUSE (2002) , Varenna, pp. 185–190 (2002)
Greco, S., Matarazzo, B., Slowinski, R.: Bipolar Sugeno and Choquet integrals. In: Proc. EUROFUSE (2002) ,Varenna (2002)
Imaoka, H.: On a subjective evaluation model by a generalized fuzzy integral. Int. J. Uncertainty, Fuzziness, and Knowledge-Based Systems 5, 517–529 (1997)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic, Studia Logica Library, vol. 8. Kluwer Acad. Publishers, Dordrecht (2000)
Klement, E.P., Mesiar, R., Pap, E.: Measure-based aggregation operators. Fuzzy Sets and System 142, 3–14 (2004)
Klir, G.J., Folger, T.A.: Fuzzy Sets. Uncertainty, and Infornation. Prentice-Hall, Englewood Cliffs (1988)
Kolesárová, A., Komorníková, M.: Triangular norm-based iterative compensatory operators. Fuzzy Sets and Systems 104, 109–120 (1999)
Marichal, J.-L.: On Sugeno integrals as an aggregation function. Fuzzy Sets and Systems 114, 347–365 (2000)
Mesiar, R.: Choquet-like integrals. J. Math. Anal. Appl. 194, 477–488 (1995)
Narukawa, Y., Torra, V.: Generalization of twofold integral and its interpretation. In: Proc. EUSFLAT (2003) , Zittau, pp. 718–722 (2003)
Pap, E.: Null-Additive Set functions. Kluwer Acad. Publ, Dordrecht (1995)
Pap, E. (ed.): Handbook of Measure Theory. Elsevier Science, Amsterdam (2002)
Schmeidler, D.: Integral representation without additivity. Proc. Amer. Math. Soc. 97, 255–261 (1986)
Shilkret, V.: Maxitive measures and integration. Indag. Math. 33, 109–116 (1971)
Šipoš, J.: Integral with respect to premeasure. Math. Slovaca 29, 141–145 (1979)
Sugeno, M.: Theory of fuzzy integrals and applications. Ph.D. doctoral dissertation, Tokyo Inst. of Tech. (1974)
Tarski, A.: Une contribution a la théorie de la mesure. Fund. Math. 15, 42–50 (1930)
Torra, V.: Twofold integral: A Choquet integral and Sugeno integral generalization. Butlletí de l’Associació Catalana d’Intel·ligència Artificial 29, 13-20 (in Catalan). Preliminary version: IIIA Research Report TR-2003-08, in English (2003)
Vitali, G.: Sulla definizione di integrale delle funzioni di una variabile. Ann. Mat. Pura ed Appl. IV 2, 111–121 (1925) ; English translation: On the definition of integral of one variable. Rivista di Matematica per le scienze sociali 20, 159–168 (1997)
Wang, Z., Klir, G.: Fuzzy measure Theory. Plenum Press, New York (1992)
Weber, S.: Two integrals and some modified versions: critical remarks. Fuzzy Sets and Systems 20, 97–105 (1986)
Zadeh, L., Probability, A.: measure of fuzzy events. J. Math. Anal. Appl. 23, 421–427 (1968)
Zadeh, L., Fuzzy, A.: sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1, 3–28 (1978)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mesiar, R., Mesiarová, A. (2004). Fuzzy Integrals. In: Torra, V., Narukawa, Y. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2004. Lecture Notes in Computer Science(), vol 3131. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27774-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-27774-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22555-3
Online ISBN: 978-3-540-27774-3
eBook Packages: Springer Book Archive