Abstract
Logical vector-valued Choquet integral models are vector-valued functions calculated by m times Choquet integral calculations with respect to the m-th set functions to the interval [0,1]. By placing restrictions on the set functions, we can get some good properties, such as a normalized output. To introduce a symmetric difference expression, some set functions are transformed into monotone set functions, and they can be interpreted by using fuzzy measure tools such as Shapley values. Similarly, we introduce a vector-valued Choquet integral for bi-capacities and their symmetric difference expressions. Despite from the vector-valued Choquet integral for set functions, the output values match with original and symmetric difference expressions.
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
Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1954)
Murofushi, T., Sugeno, M.: A theory of fuzzy measure: Representation, the Choquet integral and null sets. J. Math. Anal. Appl. 159, 532–549 (1991)
Grabisch, M.: k-Order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92, 167–189 (1999)
Takahagi, E.: Fuzzy three-valued switching functions using Choquet integral. Journal of Advanced Computational Intelligence and Intelligent Informatics 7(1), 47–52 (2003)
Takahagi, E.: Fuzzy Integral Based Fuzzy Switching Functions. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 129–150. Springer, Heidelberg (2004)
Grabisch, M., Labreuche, C.: Bi-capacities — Part I: Definition, Möbius transform and interaction. Fuzzy Sets and Systems 151, 211–236 (2005)
Grabisch, M., Labreuche, C.: Bi-capacities — Part II: The Choquet integral. Fuzzy Sets and Systems 151, 237–259 (2005)
Grabisch, M., Lange, F.: Interaction transform for bi-set functions over a finite set. Information Sciences 176(16), 2279–2303 (2006)
Fujimoto, K., Murofushi, T., Sugeno, M.: k-additivity and C-decomposability of bi-capacities and its integral. Fuzzy Sets and Systems 158, 1698–1712 (2007)
Grabisch, M., Labreuche, C.: Bipolarization of posets and natural interpolation. Journal of Mathematical Analysis and Applications 343(2), 1080–1097 (2008)
Takahagi, E.: A Choquet integral model with multiple outputs and its application to classifications. Journal of Japan Society for Fuzzy Theory and Intelligent Informatics 22(4), 481–484 (2010) (in Japanese)
Takahagi, E.: Vector-valued Choquet integral models: Relations among set functions and properties. Journal of Japan Society for Fuzzy Theory and Intelligent Informatics 23(4), 596–603 (2011) (in Japanese)
Takahagi, E.: Multiple-output Choquet integral models and their applications in classification methods. International Journal of Intelligent Technologies and Applied Statistics 4(4), 519–530 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Takahagi, E. (2012). Vector-Valued Choquet Integrals for Set Functions and Bi-capacities. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31724-8_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-31724-8_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31723-1
Online ISBN: 978-3-642-31724-8
eBook Packages: Computer ScienceComputer Science (R0)