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Aggregation functions: Means ; Marichal, Jean-Luc ; et al in Information Sciences (2011), 181(1), 1-22 This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic ... [more ▼] This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means). [less ▲] Detailed reference viewed: 107 (0 UL)Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes ; Marichal, Jean-Luc ; et al in Information Sciences (2011), 181(1), 23-43 In this second part of our state-of-the-art overview on aggregation theory, based again on our recent monograph on aggregation functions, we focus on several construction methods for aggregation functions ... [more ▼] In this second part of our state-of-the-art overview on aggregation theory, based again on our recent monograph on aggregation functions, we focus on several construction methods for aggregation functions and on special classes of aggregation functions, covering the well-known conjunctive, disjunctive, and mixed aggregation functions. Some fields of applications are included. [less ▲] Detailed reference viewed: 119 (2 UL)Axiomatizations of the discrete Choquet integral and extensions Couceiro, Miguel ; Marichal, Jean-Luc in Galichet, Sylvie; Montero, Javier; Mauris, Gilles (Eds.) Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011) and LFA-2011 (2011) Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the ... [more ▼] Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, the latter functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. We also propose a simultaneous generalization of horizontal min-additivity and horizontal max-additivity, called horizontal median-additivity, and we describe the corresponding function class. Additional conditions then reduce this class to that of symmetric Lovász extensions, which includes the discrete symmetric Choquet integrals. [less ▲] Detailed reference viewed: 86 (1 UL)Weighted Banzhaf power and interaction indexes through weighted approximations of games Marichal, Jean-Luc ; Mathonet, Pierre in Dubois, Didier; Grabisch, Michel; Mesiar, Radko (Eds.) et al 32nd Linz Seminar on Fuzzy Set Theory (LINZ 2011) - Decision Theory: Qualitative and Quantitative Approaches (2011) In cooperative game theory, various kinds of power indexes are used to measure the influence that a given player has on the outcome of the game or to define a way of sharing the benefits of the game among ... [more ▼] In cooperative game theory, various kinds of power indexes are used to measure the influence that a given player has on the outcome of the game or to define a way of sharing the benefits of the game among the players. The best known power indexes are due to Shapley [15,16] and Banzhaf [1,5] and there are many other examples of such indexes in the literature. When one is concerned by the analysis of the behavior of players in a game, the information provided by power indexes might be far insufficient, for instance due to the lack of information on how the players interact within the game. The notion of interaction index was then introduced to measure an interaction degree among players in coalitions; see [13,12,7,8,14,10,6] for the definitions and axiomatic characterizations of the Shapley and Banzhaf interaction indexes as well as many others. In addition to the axiomatic characterizations the Shapley power index and the Banzhaf power and interaction indexes were shown to be solutions of simple least squares approximation problems (see [2] for the Shapley index, [11] for the Banzhaf power index and [9] for the Banzhaf interaction index). We generalize the non-weighted approach of [11,9] by adding a weighted, probabilistic viewpoint: A weight w(S) is assigned to every coalition S of players that represents the probability that coalition S forms. The solution of the weighted least squares problem associated with the probability distribution w was given in [3,4] in the special case when the players behave independently of each other to form coalitions. In this particular setting we introduce a weighted Banzhaf interaction index associated with w by considering, as in [11,9], the leading coefficients of the approximations of the game by polynomials of specified degrees.We then study the most important properties of these weighted indexes and their relations with the classical Banzhaf and Shapley indexes. [less ▲] Detailed reference viewed: 79 (0 UL)On three properties of the discrete Choquet integral ; Marichal, Jean-Luc in Dubois, Didier; Grabisch, Michel; Mesiar, Radko (Eds.) et al 32nd Linz Seminar on Fuzzy Set Theory (LINZ 2011) - Decision Theory: Qualitative and Quantitative Approaches (2011) Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the ... [more ▼] Three important properties in aggregation theory are investigated, namely horizontal min-additivity, horizontal max-additivity, and comonotonic additivity, which are defined by certain relaxations of the Cauchy functional equation in several variables. We show that these properties are equivalent and we completely describe the functions characterized by them. By adding some regularity conditions, these functions coincide with the Lovász extensions vanishing at the origin, which subsume the discrete Choquet integrals. [less ▲] Detailed reference viewed: 34 (0 UL)Quasi-polynomial functions over bounded distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2010), 80(3), 319-334 In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p ... [more ▼] In [6] the authors introduced the notion of quasi-polynomial function as being a mapping $f\colon X^n\to X$ defined and valued on a bounded chain $X$ and which can be factorized as $f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))$, where $p$ is a polynomial function (i.e., a combination of variables and constants using the chain operations $\wedge$ and $\vee$) and $\varphi$ is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions. [less ▲] Detailed reference viewed: 106 (2 UL)Measuring the influence of the kth largest variable on functions over the unit hypercube Marichal, Jean-Luc ; Mathonet, Pierre in Torra, Vicenc; Narukawa, Yasuo; Daumas, Marc (Eds.) Modeling Decisions for Artificial Intelligence: Proceedings 7th International Conference, MDAI 2010, Perpignan, France, October 27-29, 2010 (2010, October 19) By considering a least squares approximation of a given square integrable function $f\colon [0,1]^n\to\R$ by a shifted $L$-statistic function (a shifted linear combination of order statistics), we define ... [more ▼] By considering a least squares approximation of a given square integrable function $f\colon [0,1]^n\to\R$ by a shifted $L$-statistic function (a shifted linear combination of order statistics), we define an index which measures the global influence of the $k$th largest variable on $f$. We show that this influence index has appealing properties and we interpret it as an average value of the difference quotient of $f$ in the direction of the $k$th largest variable or, under certain natural conditions on $f$, as an average value of the derivative of $f$ in the direction of the $k$th largest variable. We also discuss a few applications of this index in statistics and aggregation theory. [less ▲] Detailed reference viewed: 63 (1 UL)Measuring the interactions among variables of functions over the unit hypercube Marichal, Jean-Luc ; Mathonet, Pierre in Torra, Vicenc; Narukawa, Yasuo; Daumas, Marc (Eds.) Modeling Decisions for Artificial Intelligence: Proceedings 7th International Conference, MDAI 2010, Perpignan, France, October 27-29, 2010 (2010, October 19) By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall ... [more ▼] By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of $f$. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of $f$ or, under certain natural conditions on $f$, as an expected value of the derivatives of $f$. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index. [less ▲] Detailed reference viewed: 57 (0 UL)Explicit descriptions of associative Sugeno integrals Couceiro, Miguel ; Marichal, Jean-Luc in Hüllermeier, Eyke; Kruse, Rudolf; Hoffmann, Frank (Eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems: 13th International Conference, IPMU 2010, Dortmund, Germany, June 28–July 2, 2010. Proceedings, Part I (2010, June 17) The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity $n\geq 1$ as well as to functions of multiple arities. In this paper, we ... [more ▼] The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity $n\geq 1$ as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of Sugeno integrals over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same. [less ▲] Detailed reference viewed: 83 (1 UL)Solving Chisini's functional equation Marichal, Jean-Luc Scientific Conference (2010, June) Detailed reference viewed: 70 (5 UL)On two generalizations of associativity Couceiro, Miguel ; Marichal, Jean-Luc Scientific Conference (2010, June) Detailed reference viewed: 40 (12 UL)Solving Chisini's functional equation Marichal, Jean-Luc in Aequationes Mathematicae (2010), 79(3), 237-260 We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x),...,G(x)), where F is a given function of n real variables. We provide necessary and ... [more ▼] We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x),...,G(x)), 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. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results. [less ▲] Detailed reference viewed: 89 (10 UL)Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices Couceiro, Miguel ; Marichal, Jean-Luc in Fuzzy Sets & Systems (2010), 161(5), 694-707 We discuss several characterizations of discrete Sugeno integrals over bounded distributive lattices as particular cases of lattice polynomial functions, that is, functions which can be represented in the ... [more ▼] We discuss several characterizations of discrete Sugeno integrals over bounded distributive lattices as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted infimum and supremum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions. [less ▲] Detailed reference viewed: 91 (7 UL)Weighted Banzhaf interaction indexes and weighted least squares Marichal, Jean-Luc ; Mathonet, Pierre Scientific Conference (2010, January) Detailed reference viewed: 25 (0 UL)Measuring the interactions among variables of functions over the unit hypercube Marichal, Jean-Luc ; Mathonet, Pierre Scientific Conference (2010, January) Detailed reference viewed: 69 (1 UL)Representations and characterizations of polynomial functions on chains Couceiro, Miguel ; Marichal, Jean-Luc in Journal of Multiple-Valued Logic & Soft Computing (2010), 16(1-2), 65-86 We are interested in representations and characterizations of lattice polynomial functions f: L^n --> L, where L is a given bounded distributive lattice. In companion papers [5,6], we investigated certain ... [more ▼] We are interested in representations and characterizations of lattice polynomial functions f: L^n --> L, where L is a given bounded distributive lattice. In companion papers [5,6], 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 [5,6] and by considering further conditions, namely comonotonic minitivity and maxitivity. [less ▲] Detailed reference viewed: 50 (5 UL)Axiomatizations of quasi-polynomial functions on bounded chains Couceiro, Miguel ; Marichal, Jean-Luc in Aequationes Mathematicae (2009), 78(1-2), 195-213 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 completely describe 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 give 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 provide characterizations accordingly. [less ▲] Detailed reference viewed: 66 (4 UL)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: 75 (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: 93 (1 UL) |
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