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On the moments and distribution of discrete Choquet integrals from continuous distributions ; Marichal, Jean-Luc in Journal of Computational & Applied Mathematics (2009), 230(1), 83-94 We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral ... [more ▼] We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [1] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions. [less ▲] Detailed reference viewed: 51 (1 UL)Distribution functions of linear combinations of lattice polynomials from the uniform distribution Marichal, Jean-Luc ; in Statistics & Probability Letters (2008), 78(8), 985-991 We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which ... [more ▼] We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovász extensions. [less ▲] Detailed reference viewed: 37 (2 UL)On the moments and the distribution of the Choquet integral ; Marichal, Jean-Luc in Štěpnička, Martin; Novák, Vilém; Bodenhofer, Ulrich (Eds.) New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, 2007. Volume I: Invited Lectures and Special Sessions (2007, September) We investigate the distribution functions and the moments of the so-called Choquet integral, also known as the Lovász extension, when regarded as a real function of a random sample drawn from a continuous ... [more ▼] We investigate the distribution functions and the moments of the so-called Choquet integral, also known as the Lovász extension, when regarded as a real function of a random sample drawn from a continuous population. Since the Choquet integral includes weighted arithmetic means, ordered weighted averaging operators, and lattice polynomials as particular cases, our results encompass the corresponding results for these aggregation operators. After recalling the results obtained by the authors in the uniform case, we present approaches that can be used in the non-uniform case to obtain moment approximations. [less ▲] Detailed reference viewed: 32 (1 UL)Entropy of bi-capacities ; Marichal, Jean-Luc in European Journal of Operational Research (2007), 178(1), 168-184 In the context of multicriteria decision making whose aggregation process is based on the Choquet integral, bi-capacities can be regarded as a natural extension of capacities when the underlying ... [more ▼] In the context of multicriteria decision making whose aggregation process is based on the Choquet integral, bi-capacities can be regarded as a natural extension of capacities when the underlying evaluation scale is bipolar. The notion of entropy, recently generalized to capacities to measure their uniformity, is now extended to bi-capacities. We show that the resulting entropy measure has a very natural interpretation in terms of the Choquet integral and satisfies many natural properties that one would expect from an entropy measure. [less ▲] Detailed reference viewed: 44 (4 UL)Axiomatic characterizations of generalized values Marichal, Jean-Luc ; ; in Discrete Applied Mathematics (2007), 155(1), 26-43 In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games ... [more ▼] In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games. Axiomatizations of two classes of generalized values, namely probabilistic generalized values and generalized semivalues, which extend probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural extensions of axioms involved in the axiomatizations of values. In the second half of the paper, special instances of generalized semivalues are also axiomatized. [less ▲] Detailed reference viewed: 54 (2 UL)Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices ; ; Marichal, Jean-Luc in Games and Economic Behavior (2006), 55(1), 72-99 In the framework of cooperative game theory, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to measure the interaction phenomena among ... [more ▼] In the framework of cooperative game theory, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to measure the interaction phenomena among players. Axiomatizations of two classes of interaction indices, namely probabilistic interaction indices and cardinal-probabilistic interaction indices, generalizing probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural generalizations of axioms involved in the axiomatizations of values. In the second half of the paper, existing instances of cardinal-probabilistic interaction indices encountered thus far in the literature are also axiomatized. [less ▲] Detailed reference viewed: 56 (1 UL)Entropy of bi-capacities ; Marichal, Jean-Luc in Montseny, Eduard; Sobrevilla, Pilar (Eds.) Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology and the 11th Rencontres Francophones sur la Logique Floue et ses Applications, Barcelona, Spain, September 7-9, 2005. (2005, September) The notion of Shannon entropy, recently generalized to capacities, is extended to bi-capacities and its main properties are studied. Detailed reference viewed: 44 (3 UL)An extension of the Shannon entropy to bi-capacities ; Marichal, Jean-Luc in Proc. 3rd Int. Summer School on Aggregation Operators and their Applications (AGOP 2005), Lugano, Switzerland, July 10-15, 2005 (2005, July) The notion of Shannon entropy, recently generalized to capacities, is extended to bi-capacities and its main properties are studied. Detailed reference viewed: 36 (1 UL)An axiomatic approach to the definition of the entropy of a discrete Choquet capacity ; Marichal, Jean-Luc ; in Information Sciences (2005), 172(1-2), 131-153 To extend the classical Shannon entropy to non-additive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this ... [more ▼] To extend the classical Shannon entropy to non-additive measures, Marichal recently introduced the concept of generalized entropy for discrete Choquet capacities. We provide a first axiomatization of this new concept on the basis of three axioms: the symmetry property, a boundary condition for which the entropy reduces to the Shannon entropy, and a generalized version of the well-known recursivity property. We also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy-like measure. Lastly, we provide an interpretation of it in the framework of aggregation by the discrete Choquet integral. [less ▲] Detailed reference viewed: 164 (5 UL)Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices ; ; Marichal, Jean-Luc in Proc. 10th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2004), Perugia, Italy, July 4-9, 2004 (2004, July) In the framework of cooperative game theory and multicriteria decision making, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to ... [more ▼] In the framework of cooperative game theory and multicriteria decision making, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to measure the interaction phenomena among players or criteria. Axiomatizations of two classes of interaction indices, namely probabilistic interaction indices and cardinal-probabilistic interaction indices, generalizing probabilistic values and semivalues, respectively, are first proposed. Three existing instances of cardinal-probabilistic interaction indices encountered thus far in the literature are also axiomatized. [less ▲] Detailed reference viewed: 64 (1 UL)An axiomatic approach to the definition of the entropy of a discrete Choquet capacity ; Marichal, Jean-Luc ; in Proc. 9th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2002), Annecy, France, July 1-5, 2002 (2002, July) Detailed reference viewed: 172 (1 UL) |
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