11 research outputs found

    Estructura Combinatoria de Politopos asociados a Medidas Difusas

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    Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 23-11-2020This PhD thesis is devoted to the study of geometric and combinatorial aspects of polytopes associated to fuzzy measures. Fuzzy measures are an essential tool, since they generalize the concept of probability. This greater generality allows applications to be developed in various elds, from the Decision Theory to the Game Theory. The set formed by all fuzzy measures on a referential set is a polytope. In the same way, many of the most relevant subfamilies of fuzzy measures are also polytopes. Studying the combinatorial structure of these polytopes arises as a natural problem that allows us to better understand the properties of the associated fuzzy measures. Knowing the combinatorial structure of these polytopes helps us to develop algorithms to generate points uniformly at random inside these polytopes. Generating points uniformly inside a polytope is a complex problem from both a theoretical and a computational point of view. Having algorithms that allow us to sample uniformly in polytopes associated to fuzzy measures allows us to solve many problems, among them the identi cation problem, i.e. estimate the fuzzy measure that underlies an observed data set...La presente tesis doctoral esta dedicada al estudio de distintas propiedades geometricas y combinatorias de politopos de medidas difusas. Las medidas difusas son una herramienta esencial puesto que generalizan el concepto de probabilidad. Esta mayor generalidad permite desarrollar aplicaciones en diversos campos, desde la Teoría de la Decision a laTeoría de Juegos. El conjunto formado por todas las medidas difusas sobre un referencial tiene estructura de politopo. De la misma forma, la mayora de las subfamilias mas relevantes de medidas difusas son tambien politopos. Estudiar la estructura combinatoria de estos politopos surge como un problema natural que nos permite comprender mejor las propiedades delas medidas difusas asociadas. Conocer la estructura combinatoria de estos politopos tambien nos ayuda a desarrollar algoritmos para generar aleatoria y uniformemente puntos dentro de estos politopos. Generar puntos de forma uniforme dentro de un politopo es un problema complejo desde el punto de vista tanto teorico como computacional. Disponer de algoritmos que nos permitan generar uniformemente en politopos asociados a medidas difusas nos permite resolver muchos problemas, entre ellos el problema de identificacion que trata de estimarla medida difusa que subyace a un conjunto de datos observado...Fac. de Ciencias MatemáticasTRUEunpu

    k-maxitive Sugeno integrals as aggregation models for ordinal preferences

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    International audienceWe consider an order variant of k-additivity, so-called k-maxitivity, and present an axiomatization of the class of k-maxitive Sugeno integrals over distributive lattices. To this goal, we characterize the class of lattice polynomial functions with degree at most k and show that k-maxitive Sugeno integrals coincide exactly with idempotent lattice polynomial functions whose degree is at most k. We also discuss the use of this parametrized notion in preference aggregation and learning. In particular, we address the question of determining optimal values of k through a case study on empirical data

    Actes des 22èmes rencontres francophones sur la Logique Floue et ses Applications, 10-11 octobre 2013, Reims, France

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    Supervised ranking : from semantics to algorithms

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    Measure and integral with purely ordinal scales

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    We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 00 on the real interval [1,1][-1,1]. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and

    Axiomatic structure of k-additive capacities

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    In this paper we deal with the problem of axiomatizing the preference relations modelled through Choquet integral with respect to a kk-additive capacity, i.e. whose Möbius transform vanishes for subsets of more than kk elements. Thus, kk-additive capacities range from probability measures (k=1k=1) to general capacities (k=nk=n). The axiomatization is done in several steps, starting from symmetric 2-additive capacities, a case related to the Gini index, and finishing with general kk-additive capacities. We put an emphasis on 2-additive capacities. Our axiomatization is done in the framework of social welfare, and complete previous results of Weymark, Gilboa and Ben Porath, and Gajdos.Axiomatic; Capacities; k-Additivity
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