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Distribution functions of linear combinations of lattice polynomials from the uniform distribution Marichal, Jean-Luc ; in Statistics & Probability Letters (2008), 78(8), 985-991 We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which ... [more ▼] We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovász extensions. [less ▲] Detailed reference viewed: 65 (2 UL)Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables Marichal, Jean-Luc in Journal of Mathematical Analysis & Applications (2008), 341(1), 200-210 By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$ \int_{]a,b[^n} f(\mathbf{x},\min x_i,\max x_i) \, d\mathbf{x}, $$ in which the minimum and the ... [more ▼] By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$ \int_{]a,b[^n} f(\mathbf{x},\min x_i,\max x_i) \, d\mathbf{x}, $$ in which the minimum and the maximum are replaced with two single variables. We demonstrate the usefulness of that expression in the computation of orness and andness average values of certain aggregation functions. By generalizing our result to Riemann-Stieltjes integrals, we also provide a method for the calculation of certain expected values and distribution functions. [less ▲] Detailed reference viewed: 87 (4 UL)Weighted lattice polynomials of independent random variables Marichal, Jean-Luc in Discrete Applied Mathematics (2008), 156(5), 685-694 We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice ... [more ▼] We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems. [less ▲] Detailed reference viewed: 69 (3 UL)Approximations of Lovász extensions and their induced interaction index Marichal, Jean-Luc ; Mathonet, Pierre in Discrete Applied Mathematics (2008), 156(1), 11-24 The Lovász extension of a pseudo-Boolean function f : {0,1}^n --> R is defined on each simplex of the standard triangulation of [0,1]^n as the unique affine function \hat f : [0,1]^n --> R that ... [more ▼] The Lovász extension of a pseudo-Boolean function f : {0,1}^n --> R is defined on each simplex of the standard triangulation of [0,1]^n as the unique affine function \hat f : [0,1]^n --> R that interpolates f at the n+1 vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses f. In this paper we investigate the least squares approximation problem of an arbitrary Lovász extension \hat f by Lovász extensions of (at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-Boolean functions was investigated by Hammer and Holzman and then solved explicitly by Grabisch, Marichal, and Roubens, giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of \hat f and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche. [less ▲] Detailed reference viewed: 69 (0 UL)Disaggregation of bipolar-valued outranking relations and application to the inference of model parameters ; Bisdorff, Raymond ; Marichal, Jean-Luc Scientific Conference (2008, January) Detailed reference viewed: 29 (3 UL)Counting non-isomorphic maximal independent sets of the n-cycle graph Bisdorff, Raymond ; Marichal, Jean-Luc in Journal of Integer Sequences (2008), 11(5), 1-16 The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent ... [more ▼] The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabeled (i.e., defined up to a rotation) maximal independent sets and the number of unlabeled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences. [less ▲] Detailed reference viewed: 80 (21 UL)Slices, slabs, and sections of the unit hypercube Marichal, Jean-Luc ; in Online Journal of Analytic Combinatorics (2008), 3 Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a half-space, or with a hyperplane of codimension 1, or with a flat ... [more ▼] Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a half-space, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Pólya's Ph.D. thesis, and we discuss several applications of these formulas. [less ▲] Detailed reference viewed: 74 (13 UL)Aggregation Functions for Multicriteria Decision Aid (invited lecture) Marichal, Jean-Luc Scientific Conference (2007, September) Detailed reference viewed: 42 (8 UL)Disaggregation of bipolar-valued outranking relations and application to the inference of model parameters ; Marichal, Jean-Luc ; Bisdorff, Raymond Scientific Conference (2007, September) Detailed reference viewed: 598 (5 UL)On the moments and the distribution of the Choquet integral ; Marichal, Jean-Luc in Štěpnička, Martin; Novák, Vilém; Bodenhofer, Ulrich (Eds.) New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, 2007. Volume I: Invited Lectures and Special Sessions (2007, September) We investigate the distribution functions and the moments of the so-called Choquet integral, also known as the Lovász extension, when regarded as a real function of a random sample drawn from a continuous ... [more ▼] We investigate the distribution functions and the moments of the so-called Choquet integral, also known as the Lovász extension, when regarded as a real function of a random sample drawn from a continuous population. Since the Choquet integral includes weighted arithmetic means, ordered weighted averaging operators, and lattice polynomials as particular cases, our results encompass the corresponding results for these aggregation operators. After recalling the results obtained by the authors in the uniform case, we present approaches that can be used in the non-uniform case to obtain moment approximations. [less ▲] Detailed reference viewed: 45 (1 UL)Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables Marichal, Jean-Luc in De Baets, Bernard; Maes, Koen C.; Mesiar, Radko (Eds.) Proc. 4th Int. Summer School on Aggregation Operators and their Applications (AGOP 2007), Ghent, Belgium, July 9-14, 2007 (2007, July) Detailed reference viewed: 66 (2 UL)Entropy of bi-capacities ; Marichal, Jean-Luc in European Journal of Operational Research (2007), 178(1), 168-184 In the context of multicriteria decision making whose aggregation process is based on the Choquet integral, bi-capacities can be regarded as a natural extension of capacities when the underlying ... [more ▼] In the context of multicriteria decision making whose aggregation process is based on the Choquet integral, bi-capacities can be regarded as a natural extension of capacities when the underlying evaluation scale is bipolar. The notion of entropy, recently generalized to capacities to measure their uniformity, is now extended to bi-capacities. We show that the resulting entropy measure has a very natural interpretation in terms of the Choquet integral and satisfies many natural properties that one would expect from an entropy measure. [less ▲] Detailed reference viewed: 51 (4 UL)Human centered processes: Toward a naturalistic decision making paradigm Bisdorff, Raymond ; ; Marichal, Jean-Luc in European Journal of Operational Research (2007), 177(3), 1313-2118 Detailed reference viewed: 77 (10 UL)k-intolerant capacities and Choquet integrals Marichal, Jean-Luc in European Journal of Operational Research (2007), 177(3), 1453-1468 We define an aggregation function to be (at most) k-intolerant if it is bounded from above by its kth lowest input value. Applying this definition to the discrete Choquet integral and its underlying ... [more ▼] We define an aggregation function to be (at most) k-intolerant if it is bounded from above by its kth lowest input value. Applying this definition to the discrete Choquet integral and its underlying capacity, we introduce the concept of k-intolerant capacities which, when varying k from 1 to n, cover all the possible capacities on n objects. Just as the concepts of k-additive capacities and p-symmetric capacities have been previously introduced essentially to overcome the problem of computational complexity of capacities, k-intolerant capacities are proposed here for the same purpose but also for dealing with intolerant or tolerant behaviors of aggregation. We also introduce axiomatically indices to appraise the extent to which a given capacity is k-intolerant and we apply them on a particular recruiting problem. [less ▲] Detailed reference viewed: 104 (3 UL)Comparison meaningful aggregation functions: a state of the art Marichal, Jean-Luc Presentation (2007, February 14) In many domains we are faced with the problem of aggregating a collection of numerical readings to obtain a mean or typical value. Such an aggregation problem is becoming more and more present in an ... [more ▼] In many domains we are faced with the problem of aggregating a collection of numerical readings to obtain a mean or typical value. Such an aggregation problem is becoming more and more present in an increasing number of areas not only of mathematics or physics, but also of engineering, economical, social, and other sciences. Various aggregation functions and processes have already been proposed in the literature and many others are still to be designed to fulfill newer and newer requirements. Studies on the aggregation problem have shown that the choice of the aggregation function is far from being arbitrary and should be based upon properties dictated by the framework in which the aggregation is performed. One of the main concerns when choosing an appropriate function is to take into account the scale types of the variables being aggregated. On this issue it was observed that the general form of the aggregation function is greatly restricted if we know the scale types of the dependent and independent variables. For instance, if all the variables define a common ordinal scale, it is clear that any relevant aggregation function cannot be constructed from usual arithmetic operations, unless these operations involve only order. Thus, computing the arithmetic mean is forbidden whereas the median or any order statistic is permitted. We present a state of the art survey on the known axiomatizations of aggregation functions mapping ordinal scales into an ordinal scale. We show that, in this ordinal context, the family of possible aggregation functions is rather poor, more or less consisting of order statistics and lattice polynomials. [less ▲] Detailed reference viewed: 24 (2 UL)Integer sequence #A127686 Marichal, Jean-Luc Diverse speeches and writings (2007) Number of non-isomorphic maximal independent sets of the n-cycle graph having no symmetry axis. Detailed reference viewed: 30 (1 UL)Integer sequence #A127683 Marichal, Jean-Luc Diverse speeches and writings (2007) Number of non-isomorphic (i.e. defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having 2n isomorphic representatives. Detailed reference viewed: 25 (2 UL)Integer sequence #A127684 Marichal, Jean-Luc Diverse speeches and writings (2007) Number of non-isomorphic (i.e. defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having n isomorphic representatives. Detailed reference viewed: 28 (1 UL)Integer sequence #A127687 Marichal, Jean-Luc Diverse speeches and writings (2007) Number of unlabeled (i.e. defined up to a rotation) maximal independent sets of the n-cycle graph. Detailed reference viewed: 24 (2 UL)Integer sequence #A127685 Marichal, Jean-Luc Diverse speeches and writings (2007) Number of non-isomorphic maximal independent sets of the n-cycle graph. Detailed reference viewed: 28 (1 UL) |
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