Aggregation-based extensions of fuzzy measures

We present a method extending fuzzy measures on N={1,...,n} (represented as Boolean utility functions) to n-ary aggregation functions (utility functions) by means of a suitable n-ary aggregation function and the Mobius transform of the considered fuzzy measure. The method generalizes the well-known Lovasz and Owen extensions of nondecreasing pseudo-Boolean functions linked to fuzzy measures. All n-ary aggregation functions suitable for the proposed construction are completely characterized, including, among others, all n-ary copulas. Associative extended aggregation functions applicable in the case of an arbitrary arity are also completely characterized.

[1]  Jean-Luc Marichal,et al.  Aggregation of interacting criteria by means of the discrete Choquet integral , 2002 .

[2]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[3]  R. Nelsen,et al.  Multivariate Archimedean Quasi-Copulas , 2002 .

[4]  R. Nelsen.  An Introduction to Copulas (Springer Series in Statistics) , 2006 .

[5]  Radko Mesiar,et al.  Ordinal sums and idempotents of copulas , 2010 .

[6]  R. Mesiar,et al.  ”Aggregation Functions”, Cambridge University Press , 2008, 2008 6th International Symposium on Intelligent Systems and Informatics.

[7]  G. Choquet.  Theory of capacities , 1954 .

[8]  Alain Chateauneuf,et al.  Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion , 1989, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[9]  Manuel Úbeda-Flores,et al.  Some new characterizations and properties of quasi-copulas , 2009, Fuzzy Sets Syst..

[10]  Fabrizio Durante,et al.  Semicopulas: characterizations and applicability , 2006, Kybernetika.

[11]  Alexander J. McNeil,et al.  Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions , 2009, 0908.3750.

[12]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[13]  Anna Kolesárová,et al.  Associative n - dimensional copulas , 2011, Kybernetika.

[14]  M. Sugeno,et al.  Fuzzy Measures and Integrals: Theory and Applications , 2000 .

[15]  Andrey G. Bronevich.  On the closure of families of fuzzy measures under eventwise aggregations , 2005, Fuzzy Sets Syst..

[16]  Jun Okamoto,et al.  Inclusion-Exclusion Integral and Its Application to Subjective Video Quality Estimation , 2010, IPMU.

[17]  G. Owen.  Multilinear Extensions of Games , 1972 .

[18]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[19]  A. Roth.  The Shapley value , 2005, Game Theory.

[20]  Gaspar Mayor,et al.  Aggregation Operators , 2002 .

[21]  Fabrizio Durante,et al.  On representations of 2-increasing binary aggregation functions , 2008, Inf. Sci..

[22]  R. Mesiar,et al.  Aggregation operators: properties, classes and construction methods , 2002 .

[23]  Radko Mesiar,et al.  2-Increasing binary aggregation operators , 2007, Inf. Sci..

[24]  Christian Genest,et al.  A Characterization of Quasi-copulas , 1999 .

[25]  Claudi Alsina,et al.  On the characterization of a class of binary operations on distribution functions , 1993 .

[26]  Anna Kolesárová,et al.  1-Lipschitz aggregation operators and quasi-copulas , 2003, Kybernetika.

[27]  R. Nelsen.  An Introduction to Copulas , 1998 .