Pivotal decompositions of functions

in Discrete Applied Mathematics (2014), 174

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary ... [more ▼]

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary section of a function only depends upon its values at two given elements. Pivotal decompositions appear to hold for various function classes, such as the class of lattice polynomial functions or the class of multilinear polynomial functions. We also define function classes characterized by pivotal decompositions and function classes characterized by their unary members and investigate links between these two concepts. [less ▲]

Preassociative aggregation functions

in Laurent, Anne; Strauss, Olivier; Bouchon-Meunier, Bernadette (Eds.) et al 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2014, Montpellier, France, July 15-19, 2014. Proceedings, Part III (2014, July 22)

We investigate the associativity property for varying-arity aggregation functions and introduce the more general property of preassociativity, a natural extension of associativity. We discuss this new ... [more ▼]

We investigate the associativity property for varying-arity aggregation functions and introduce the more general property of preassociativity, a natural extension of associativity. We discuss this new property and describe certain classes of preassociative functions. [less ▲]

Associativity, preassociativity, and string functions

Scientific Conference (2014, June 20)

Preassociativity for aggregation functions

Scientific Conference (2014, June 16)

Analyse de fiabilité des systèmes et description par des polynômes latticiels

Presentation (2014, March 07)

Subsignatures of systems

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 ▲]

Fonctions d’agrégation barycentriquement associatives

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 ▲]

Discrete integrals based on comonotonic modularity

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 ▲]

On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes

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 ▲]

On the cardinality index of fuzzy measures and the signatures of coherent systems

in Mesiar, Radko; Pap, Endre; Klement, Erich Peter (Eds.) 34th Linz Seminar on Fuzzy Set Theory (LINZ 2013) - Non-Classical Measures and Integrals (2013)