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Preassociativity for aggregation functions Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2014, June 16) Detailed reference viewed: 72 (5 UL)Fonctions d'agrégation associatives et préassociatives (invited lecture) Marichal, Jean-Luc Presentation (2014, May 20) Dans cette présentation, nous étudions la propriété d'associativité pour les fonctions d'agrégation à arités multiples et nous introduisons la propriété plus générale de préassociativité qui n'implique ... [more ▼] Dans cette présentation, nous étudions la propriété d'associativité pour les fonctions d'agrégation à arités multiples et nous introduisons la propriété plus générale de préassociativité qui n'implique aucune composition de fonctions. Nous étudions cette nouvelle propriété et nous décrivons certaines classes de fonction préassociatives. Nous montrons aussi comment des axiomatisations de plusieurs classes de functions (comme les semi-groupes aczéliens ou les polynômes latticiels associatifs) peuvent être obtenues en affaiblissant l'associativité en la remplaçant par la préassociativité. [less ▲] Detailed reference viewed: 49 (8 UL)Analyse de fiabilité des systèmes et description par des polynômes latticiels Marichal, Jean-Luc Presentation (2014, March 07) Detailed reference viewed: 48 (13 UL)Fonctions d'agrégation pour l'aide multicritère à la décision Marichal, Jean-Luc Presentation (2014, February 21) Detailed reference viewed: 99 (7 UL)Subsignatures of systems Marichal, Jean-Luc in Journal of Multivariate Analysis (2014), 124 We introduce the concept of subsignature for semicoherent systems as a class of indexes that range from the system signature to the Barlow-Proschan importance index. Specifically, given a nonempty subset ... [more ▼] We introduce the concept of subsignature for semicoherent systems as a class of indexes that range from the system signature to the Barlow-Proschan importance index. Specifically, given a nonempty subset M of the set of components of a system, we define the M-signature of the system as the |M|-tuple whose k-th coordinate is the probability that the k-th failure among the components in M causes the system to fail. We give various explicit linear expressions for this probability in terms of the structure function and the distribution of the component lifetimes. We also examine the case of exchangeable lifetimes and the special case when the lifetime are i.i.d. and M is a modular set. [less ▲] Detailed reference viewed: 83 (20 UL)Fonctions associatives et préassociatives Marichal, Jean-Luc ; Teheux, Bruno Scientific Conference (2013, October 18) We investigate the associativity property for functions of multiple arities and introduce the more general property of preassociativity, an extension of associativity which does not involve any ... [more ▼] We investigate the associativity property for functions of multiple arities and introduce the more general property of preassociativity, an extension of associativity which does not involve any composition of functions. We discuss this new property and describe certain classes of preassociative functions. We also show how axiomatizations of several function classes can be extended by relaxing associativity to preassociativity. [less ▲] Detailed reference viewed: 56 (9 UL)Actes des 22èmes rencontres francophones sur la Logique Floue et ses Applications, 10-11 octobre 2013, Reims, France Marichal, Jean-Luc ; ; Book published by Université de Reims Champagne-Ardenne (2013) Detailed reference viewed: 160 (7 UL)Fonctions d’agrégation barycentriquement associatives Marichal, Jean-Luc ; Teheux, Bruno in Marichal, Jean-Luc; Essounbouli, Najib; Guelton, Kevin (Eds.) Actes des 22èmes rencontres francophones sur la Logique Floue et ses Applications, 10-11 octobre 2013, Reims, France (2013, October) We investigate the algebraic property of barycentric associativity for aggregation functions. This property, well-known in Kolmogoroff-Nagumo’s axiomatization of the quasi-arithmetic means, is often ... [more ▼] We investigate the algebraic property of barycentric associativity for aggregation functions. This property, well-known in Kolmogoroff-Nagumo’s axiomatization of the quasi-arithmetic means, is often considered as very natural whenever the aggregation process is of an (arithmetic, geometric, harmonic...) mean type. We recall the definition of this property and propose some extensions. We also present some results, some rather surprising, related to these properties. [less ▲] Detailed reference viewed: 135 (11 UL)Discrete integrals based on comonotonic modularity ; Marichal, Jean-Luc in Axioms (2013), 2(3), 390-403 It is known that several discrete integrals, including the Choquet and Sugeno integrals as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of ... [more ▼] It is known that several discrete integrals, including the Choquet and Sugeno integrals as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families of discrete integrals that are comonotonically modular, including signed Choquet integrals and symmetric signed Choquet integrals as well as natural extensions of Sugeno integrals. [less ▲] Detailed reference viewed: 102 (5 UL)Computing system signatures through reliability functions Marichal, Jean-Luc ; Mathonet, Pierre in Statistics & Probability Letters (2013), 83(3), 710-717 It is known that the Barlow-Proschan index of a system with i.i.d. component lifetimes coincides with the Shapley value, a concept introduced earlier in cooperative game theory. Due to a result by Owen ... [more ▼] It is known that the Barlow-Proschan index of a system with i.i.d. component lifetimes coincides with the Shapley value, a concept introduced earlier in cooperative game theory. Due to a result by Owen, this index can be computed efficiently by integrating the first derivatives of the reliability function of the system along the main diagonal of the unit hypercube. The Samaniego signature of such a system is another important index that can be computed for instance by Boland's formula, which requires the knowledge of every value of the associated structure function. We show how the signature can be computed more efficiently from the diagonal section of the reliability function via derivatives. We then apply our method to the computation of signatures for systems partitioned into disjoint modules with known signatures. [less ▲] Detailed reference viewed: 152 (21 UL)On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes Marichal, Jean-Luc ; Mathonet, Pierre in Journal of Multivariate Analysis (2013), 115 For a coherent system the Barlow-Proschan importance index, defined when the component lifetimes are independent, measures the probability that the failure of a given component causes the system to fail ... [more ▼] For a coherent system the Barlow-Proschan importance index, defined when the component lifetimes are independent, measures the probability that the failure of a given component causes the system to fail. Iyer (1992) extended this concept to the more general case when the component lifetimes are jointly absolutely continuous but not necessarily independent. Assuming only that the joint distribution of component lifetimes has no ties, we give an explicit expression for this extended index in terms of the discrete derivatives of the structure function and provide an interpretation of it as a probabilistic value, a concept introduced in game theory. This enables us to interpret Iyer's formula in this more general setting. We also discuss the analogy between this concept and that of system signature and show how it can be used to define a symmetry index for systems. [less ▲] Detailed reference viewed: 143 (14 UL)On comonotonically modular functions ; Marichal, Jean-Luc in Mesiar, Radko; Pap, Endre; Klement, Erich Peter (Eds.) 34th Linz Seminar on Fuzzy Set Theory (LINZ 2013) - Non-Classical Measures and Integrals (2013) Detailed reference viewed: 24 (2 UL)Pivotal decompositions of aggregation functions Marichal, Jean-Luc ; Teheux, Bruno in Mesiar, Radko; Pap, Endre; Klement, Erich Peter (Eds.) 34th Linz Seminar on Fuzzy Set Theory (LINZ 2013) - Non-Classical Measures and Integrals (2013) Detailed reference viewed: 44 (11 UL)On the cardinality index of fuzzy measures and the signatures of coherent systems Mathonet, Pierre ; Marichal, Jean-Luc in Mesiar, Radko; Pap, Endre; Klement, Erich Peter (Eds.) 34th Linz Seminar on Fuzzy Set Theory (LINZ 2013) - Non-Classical Measures and Integrals (2013) Detailed reference viewed: 59 (1 UL)Quasi-extensions de Lovász et leur version symétrique ; Marichal, Jean-Luc in Rencontres francophones sur la logique floue et ses applications 2012 (2012, November 15) We present a study of the class of quasi-Lovász extensions (i.e. functions which are a composition of a Lovász extension with a nondecreasing function vanishing at the origin) as well as that of their ... [more ▼] We present a study of the class of quasi-Lovász extensions (i.e. functions which are a composition of a Lovász extension with a nondecreasing function vanishing at the origin) as well as that of their symmetric variants. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals and symmetric discrete Choquet integrals, respectively, whose variables are transformed by a given utility function. [less ▲] Detailed reference viewed: 37 (0 UL)A classification of bisymmetric polynomial functions over integral domains of characteristic zero Marichal, Jean-Luc ; Mathonet, Pierre in Aequationes Mathematicae (2012), 84(1-2), 125-136 We describe the class of n-variable polynomial functions that satisfy Aczél's bisymmetry property over an arbitrary integral domain of characteristic zero with identity. Detailed reference viewed: 123 (9 UL)Aczélian n-ary semigroups Couceiro, Miguel ; Marichal, Jean-Luc in Semigroup Forum (2012), 85(1), 81-90 We show that the real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczél ... [more ▼] We show that the real continuous, symmetric, and cancellative n-ary semigroups are topologically order-isomorphic to additive real n-ary semigroups. The binary case (n=2) was originally proved by Aczél (1949); there symmetry was redundant. [less ▲] Detailed reference viewed: 133 (18 UL)Symmetric approximations of pseudo-Boolean functions with applications to influence indexes Marichal, Jean-Luc ; Mathonet, Pierre in Applied Mathematics Letters (2012), 25(8), 1121-1126 We introduce an index for measuring the influence of the $k$th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear ... [more ▼] We introduce an index for measuring the influence of the $k$th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making. [less ▲] Detailed reference viewed: 66 (4 UL)Locally monotone Boolean and pseudo-Boolean functions Couceiro, Miguel ; Marichal, Jean-Luc ; Waldhauser, Tamás in Discrete Applied Mathematics (2012), 160(12), 1651-1660 We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on ... [more ▼] We propose local versions of monotonicity for Boolean and pseudo-Boolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are shown to be tightly related to lattice counterparts of classical partial derivatives via the notion of permutable derivatives. More precisely, p-locally monotone functions are shown to have p-permutable lattice derivatives and, in the case of symmetric functions, these two notions coincide. We provide further results relating these two notions, and present a classification of p-locally monotone functions, as well as of functions having p-permutable derivatives, in terms of certain forbidden "sections", i.e., functions which can be obtained by substituting constants for variables. This description is made explicit in the special case when p=2. [less ▲] Detailed reference viewed: 93 (3 UL)Reliability of systems with dependent components based on lattice polynomial description ; Marichal, Jean-Luc in Stochastic Models (2012), 28(1), 167-184 Reliability of a system is considered where the components' random lifetimes may be dependent. The structure of the system is described by an associated "lattice polynomial" function. Based on that ... [more ▼] Reliability of a system is considered where the components' random lifetimes may be dependent. The structure of the system is described by an associated "lattice polynomial" function. Based on that descriptor, general framework formulas are developed and used to obtain direct results for the cases where a) the lifetimes are "Bayes-dependent", that is, their interdependence is due to external factors (in particular, where the factor is the "preliminary phase" duration) and b) where the lifetimes' dependence is implied by upper or lower bounds on lifetimes of components in some subsets of the system. (The bounds may be imposed externally based, say, on the connections environment.) Several special cases are investigated in detail. [less ▲] Detailed reference viewed: 90 (6 UL) |
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