Grabisch, Michel and Miranda Menéndez, Pedro
(2007)
*On the k-additive Core of Capacities.*
In
New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, 2007.
University of Ostrava, Ostrava, pp. 257-263.
ISBN 978-80-7368-386-3

PDF
Restricted to Repository staff only until 2020. 163kB |

Official URL: http://www.eusflat.org/proceedings/EUSFLAT_2007/papers/Michel_Grabisch_(138).pdf

## Abstract

We investigate in this paper the set of k-

additive capacities dominating a given capacity,

which we call the k-additive core. We

study its structure through achievable families,

which play the role of maximal chains

in the classical case (k = 1), and show that

associated capacities are elements (possibly a

vertex) of the k-additive core when the capacity

is (k+1)-monotone. As a particular case,

we study the set of k-additive belief functions

dominating a belief function. The problem

of finding all vertices of the k-additive core is still an open question.

Item Type: | Book Section |
---|---|

Uncontrolled Keywords: | k-additive capacity, core, belief function |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17056 |

References: | J.-P. Barthélemy. Monotone functions on finite lattices: an ordinal approach to capacities, belief and necessity functions. In J. Fodor, B. De Baets, and P. Perny, editors, Preferences and Decisions under Incomplete Knowledge,pages 195–208. Physica Verlag, 2000. A. Chateauneuf and J.-Y. Jaffray. Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences, 17:263–283, 1989. G. Choquet. Theory of capacities. Annales de l’Institut Fourier, 5:131–295, 1953. A. P. Dempster. Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist., 38:325–339, 1967. M. Grabisch. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, 92:167–189, 1997. M. Grabisch. On lower and upper approximation of fuzzy measures by k-order additive measures. In 7th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98), pages 1577–1584, Paris, France, July 1998. M. Grabisch. Upper approximation of non-additive measures by k-additive measures — the case of belief functions. In Proc. of the 1st Int. Symp. on Imprecise Probabilities and Their Applications, Ghent, Belgium, June 1999. M. Grabisch. On upper approximation of fuzzy measures by k-order additive measures. In B. Bouchon-Meunier, R. R. Yager, and L. Zadeh, editors, Information, Uncertainty, Fusion, pages 105–118. Kluwer Scientific Publ., 2000. M. Grabisch and P. Miranda. A study of the k-additive core of capacities through achievable families. In SCIS-ISIS 2006, 3nd Int. Conf. on Soft Computing and Intelligent Systems and 7th Int. Symp. on Advanced Intelligent Systems, Yokohama, Japan, September 2006. T. Ichiishi. Super-modularity: applications to convex games and to the greedy algorithm for LP. J. Econom. Theory, 25:283–286, 1981. P. Miranda, M. Grabisch, and P. Gil. Dominance of fuzzy measures by k-additive belief functions. In Int.Fuzzy Systems Association World Congress (IFSA 2003), pages 143–146, Istanbul, Turkey, June 2003. G. Shafer. A Mathematical Theory of Evidence. Princeton Univ. Press, 1976. L. S. Shapley. Core of convex games. Int. J. Game Theory, 1:11–26, 1971. M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974. P. Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991 |

Deposited On: | 13 Nov 2012 10:25 |

Last Modified: | 07 Feb 2014 09:41 |

Repository Staff Only: item control page