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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: 78 (1 UL)Aggregation functions ; Marichal, Jean-Luc ; et al Book published by Cambridge University Press (2009) Aggregation is the process of combining several numerical values into a single representative value, and an aggregation function performs this operation. These functions arise wherever aggregating ... [more ▼] Aggregation is the process of combining several numerical values into a single representative value, and an aggregation function performs this operation. These functions arise wherever aggregating information is important: applied and pure mathematics (probability, statistics, decision theory, functional equations), operations research, computer science, and many applied fields (economics and finance, pattern recognition and image processing, data fusion, etc.). This is a comprehensive, rigorous and self-contained exposition of aggregation functions. Classes of aggregation functions covered include triangular norms and conorms, copulas, means and averages, and those based on nonadditive integrals. The properties of each method, as well as their interpretation and analysis, are studied in depth, together with construction methods and practical identification methods. Special attention is given to the nature of scales on which values to be aggregated are defined (ordinal, interval, ratio, bipolar). It is an ideal introduction for graduate students and a unique resource for researchers. [less ▲] Detailed reference viewed: 112 (8 UL)Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art Marichal, Jean-Luc ; in Aequationes Mathematicae (2009), 77(3), 207-236 We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful ... [more ▼] We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and comparison meaningful functions on independent ordinal scales. It appears that the most prominent meaningful aggregation functions are lattice polynomial functions, that is, functions built only on projections and minimum and maximum operations. [less ▲] Detailed reference viewed: 99 (1 UL)Weighted lattice polynomials Marichal, Jean-Luc in Discrete Mathematics (2009), 309(4), 814-820 We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an ... [more ▼] We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a median based decomposition formula. [less ▲] Detailed reference viewed: 92 (6 UL)Analyse de fiabilité des systèmes semicohérents et description par des polynômes latticiels ; Marichal, Jean-Luc Scientific Conference (2009, February 10) Detailed reference viewed: 25 (0 UL)Behavioral analysis of aggregation functions Marichal, Jean-Luc ; ; et al in Bodenhofer, Ulrich; De Baets, Bernard; Klement, Erich Peter (Eds.) et al Proc. 30th Linz Seminar on Fuzzy Set Theory (LINZ 2009): The Legacy of 30 Seminars – Where Do We Stand and Where Do We Go? (2009, February) Detailed reference viewed: 85 (1 UL)Infinitary aggregation ; ; et al in Bodenhofer, Ulrich; De Baets, Bernard; Klement, Erich Peter (Eds.) et al Proc. 30th Linz Seminar on Fuzzy Set Theory (LINZ 2009): The Legacy of 30 Seminars – Where Do We Stand and Where Do We Go? (2009, February) In this paper, based on [12, 18], we present infinitary aggregation functions on sequences possessing some a priori given properties. General infinitary aggregation is also discussed, and the connection ... [more ▼] In this paper, based on [12, 18], we present infinitary aggregation functions on sequences possessing some a priori given properties. General infinitary aggregation is also discussed, and the connection with integrals, e.g., Lebesgue, Choquet and Sugeno integrals, is given. [less ▲] Detailed reference viewed: 46 (0 UL)Aggregation on bipolar scales ; Marichal, Jean-Luc ; et al in Bodenhofer, Ulrich; De Baets, Bernard; Klement, Erich Peter (Eds.) et al Proc. 30th Linz Seminar on Fuzzy Set Theory (LINZ 2009): The Legacy of 30 Seminars – Where Do We Stand and Where Do We Go? (2009, February) Detailed reference viewed: 84 (0 UL)Contribution on some construction methods for aggregation functions ; ; et al In this paper, based on [14], we present some well established construction methods for aggregation functions as well as some new ones. Detailed reference viewed: 36 (0 UL)Characterizations of discrete Sugeno integrals as lattice polynomial functions Couceiro, Miguel ; Marichal, Jean-Luc We survey recent characterizations of the class of lattice polynomial functions and of the subclass of discrete Sugeno integrals defined on bounded chains. Detailed reference viewed: 39 (1 UL)Quasi-polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Carvalho, J. P.; Dubois, D.; Kaymak, U. (Eds.) et al Proc. of 2009 Int. Fuzzy Systems Assoc. World Congress and 2009 Int. Conf. of the Eur. Soc. for Fuzzy Logic and Technology (IFSA-EUSFLAT 2009 Joint Conference) (2009) Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain ... [more ▼] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We present complete descriptions of the function classes axiomatized by each of these properties, up to weak versions of monotonicity, in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasipolynomial function which appears as a natural extension of the well-established notion of polynomial function. We present further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and present characterizations accordingly. [less ▲] Detailed reference viewed: 53 (0 UL)Polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Carvalho, J.-P.; Dubois, D.; Kaymak, U. (Eds.) et al Proc. of 2009 Int. Fuzzy Systems Assoc. World Congress and 2009 Int. Conf. of the Eur. Soc. for Fuzzy Logic and Technology (IFSA-EUSFLAT 2009 Joint Conference) (2009) We are interested in representations and characterizations of lattice polynomial functions $f\colon L^n\to L$, where $L$ is a given bounded distributive lattice. In an earlier paper [4,5], we investigated ... [more ▼] We are interested in representations and characterizations of lattice polynomial functions $f\colon L^n\to L$, where $L$ is a given bounded distributive lattice. In an earlier paper [4,5], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when $L$ is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [4,5] and by considering further conditions, namely comonotonic minitivity and maxitivity. [less ▲] Detailed reference viewed: 53 (0 UL)From discrete Sugeno integrals to generalized lattice polynomial functions: axiomatizations and representations (invited lecture) Couceiro, Miguel ; Marichal, Jean-Luc in González, Manuel; Mayor, Gaspar; Suner, Jaume (Eds.) et al Proc. 5th Int. Summer School on Aggregation Operators and their Applications (AGOP 2009) (2009) Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain ... [more ▼] Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We present complete descriptions of the function classes axiomatized by each of these properties, up to weak versions of monotonicity, in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We present further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. [less ▲] Detailed reference viewed: 37 (1 UL)Monograph: aggregation functions ; Marichal, Jean-Luc ; et al in Acta Polytechnica Hungarica (2009), 6(1), 79-94 There is given a short overview of the monograph "Aggregation Functions" (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap), Cambridge University Press (in press) with more details from introductory ... [more ▼] There is given a short overview of the monograph "Aggregation Functions" (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap), Cambridge University Press (in press) with more details from introductory Chapters 1 and 2. [less ▲] Detailed reference viewed: 157 (1 UL)Aggregation functions for decision making Marichal, Jean-Luc in Bouyssou, Denis; Dubois, Didier; Pirlot, Marc (Eds.) et al Decision-Making Process – Concepts and Methods (2009) Detailed reference viewed: 122 (9 UL)The Chisini mean revisited Marichal, Jean-Luc in González, Manuel; Mayor, Gaspar; Suner, Jaume (Eds.) et al Proc. 5th Int. Summer School on Aggregation Operators and their Applications (AGOP 2009) (2009) We investigate the $n$-variable real functions $\G$ that are solutions of the functional equation $\F(\bfx)=\F(\G(\bfx),\ldots,\G(\bfx))$, where $\F$ is a given function of $n$ real variables. We provide ... [more ▼] We investigate the $n$-variable real functions $\G$ that are solutions of the functional equation $\F(\bfx)=\F(\G(\bfx),\ldots,\G(\bfx))$, where $\F$ is a given function of $n$ real variables. We provide necessary and sufficient conditions on $\F$ for the existence and uniqueness of solutions. When $\F$ is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. Such solutions, called Chisini means, are then thoroughly investigated. [less ▲] Detailed reference viewed: 59 (5 UL)Reliability analysis and lattice polynomial system representation ; Marichal, Jean-Luc Scientific Conference (2008, October 17) Detailed reference viewed: 27 (0 UL)“Aggregation Functions”, Cambridge University Press ; Marichal, Jean-Luc ; et al in Proc. of the 6th Int. Symposium on Intelligent Systems and Informatics (SISY 2008) (2008, September) There is given a short overview of the monograph ”Aggregation Functions” (M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap), Cambridge University Press (in press). Detailed reference viewed: 83 (5 UL)Disaggregation of bipolar-valued outranking relations ; Marichal, Jean-Luc ; Bisdorff, Raymond in Le Thi, Hoai An; Bouvry, Pascal; Pham Dinh, Tao (Eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences (2008, August 28) In this article, we tackle the problem of exploring the structure of the data which is underlying a bipolar-valued outranking relation. More precisely, we show how the performances of alternatives and ... [more ▼] In this article, we tackle the problem of exploring the structure of the data which is underlying a bipolar-valued outranking relation. More precisely, we show how the performances of alternatives and weights related to criteria can be determined from three different formulations of the bipolar-valued outranking relations, which are given beforehand. [less ▲] Detailed reference viewed: 86 (6 UL)System reliability and weighted lattice polynomials ; Marichal, Jean-Luc in Probability in the Engineering and Informational Sciences (2008), 22(3), 373-388 The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such ... [more ▼] The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of "indicator" variables. A connection is studied between Y and order statistics of the set of arguments. [less ▲] Detailed reference viewed: 75 (4 UL) |
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