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A classification of barycentrically associative polynomial functions ; Mathonet, Pierre ; in Aequationes Mathematicae (2015), 89(5), 1281-1291 We describe the class of polynomial functions which are barycentrically associative over an infinite commutative integral domain. Detailed reference viewed: 53 (3 ULg)On modular decompositions of system signatures ; Mathonet, Pierre in Journal of Multivariate Analysis (2015), 134 Considering a semicoherent system made up of $n$ components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that ... [more ▼] Considering a semicoherent system made up of $n$ components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. This $n$-tuple, which depends only on the structure of the system and not on the distribution of the component lifetimes, is a very useful tool in the theoretical analysis of coherent systems. It was shown in two independent recent papers how the structural signature of a system partitioned into two disjoint modules can be computed from the signatures of these modules. In this work we consider the general case of a system partitioned into an arbitrary number of disjoint modules organized in an arbitrary way and we provide a general formula for the signature of the system in terms of the signatures of the modules. The concept of signature was recently extended to the general case of semicoherent systems whose components may have dependent lifetimes. The same definition for the $n$-tuple gives rise to the probability signature, which may depend on both the structure of the system and the probability distribution of the component lifetimes. In this general setting, we show how under a natural condition on the distribution of the lifetimes, the probability signature of the system can be expressed in terms of the probability signatures of the modules. We finally discuss a few situations where this condition holds in the non-i.i.d. and nonexchangeable cases and provide some applications of the main results. [less ▲] Detailed reference viewed: 25 (1 ULg)Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach ; Mathonet, Pierre in Discrete Applied Mathematics (2014), 179 The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the stan- dard least squares approximation of the function by a ... [more ▼] The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the stan- dard least squares approximation of the function by a pseudo-Boolean func- tion of a specified degree. We first observe that this property still holds if we consider approximations by pseudo-Boolean functions depending only on specified variables. We then show that the Banzhaf influence index can also be obtained from the latter approximation problem. Considering cer- tain weighted versions of this approximation problem, we introduce a class of weighted Banzhaf influence indexes, analyze their most important properties, and point out similarities between the weighted Banzhaf influence index and the corresponding weighted Banzhaf interaction index. We also discuss the issue of reconstructing a pseudo-Boolean function from prescribed influences and point out very different behaviors in the weighted and non-weighted cases. [less ▲] Detailed reference viewed: 11 (0 ULg)Computing system signatures through reliability functions ; Mathonet, Pierre in Statistics & Probability Letters (2013), 83(3), 710-717 It is known that the Barlow-Proschan index of a system with i.i.d. component lifetimes coincides with the Shapley value, a concept introduced earlier in cooperative game theory. Due to a result by Owen ... [more ▼] It is known that the Barlow-Proschan index of a system with i.i.d. component lifetimes coincides with the Shapley value, a concept introduced earlier in cooperative game theory. Due to a result by Owen, this index can be computed efficiently by integrating the first derivatives of the reliability function of the system along the main diagonal of the unit hypercube. The Samaniego signature of such a system is another important index that can be computed for instance by Boland's formula, which requires the knowledge of every value of the associated structure function. We show how the signature can be computed more efficiently from the diagonal section of the reliability function via derivatives. We then apply our method to the computation of signatures for systems partitioned into disjoint modules with known signatures. [less ▲] Detailed reference viewed: 31 (2 ULg)On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes ; Mathonet, Pierre 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 ▲] Detailed reference viewed: 26 (5 ULg)A classification of bisymmetric polynomial functions over integral domains of characteristic zero ; Mathonet, Pierre in Aequationes Mathematicae (2012), 84 We describe the class of n-variable polynomial functions that satisfy Aczel’s bisymmetry property over an arbitrary integral domain of characteristic zero with identity. Detailed reference viewed: 27 (4 ULg)Symmetric approximations of pseudo-Boolean functions with applications to influence indexes ; Mathonet, Pierre in Applied Mathematics Letters (2012), 25(8), 1121-1126 We introduce an index for measuring the influence of the $k$th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear ... [more ▼] We introduce an index for measuring the influence of the $k$th smallest variable on a pseudo-Boolean function. This index is defined from a weighted least squares approximation of the function by linear combinations of order statistic functions. We give explicit expressions for both the index and the approximation and discuss some properties of the index. Finally, we show that this index subsumes the concept of system signature in engineering reliability and that of cardinality index in decision making. [less ▲] Detailed reference viewed: 13 (0 ULg)On signature-based expressions of system reliability ; Mathonet, Pierre ; in Journal of Multivariate Analysis (2011), 102(10), 1410-1416 The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular ... [more ▼] The concept of signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. This concept proved to be useful in the analysis of theoretical behaviors of systems. In particular, it provides an interesting signature-based representation of the system reliability in terms of reliabilities of k-out-of-n systems. In the non-i.i.d. case, we show that, at any time, this representation still holds true for every coherent system if and only if the component states are exchangeable. We also discuss conditions for obtaining an alternative representation of the system reliability in which the signature is replaced by its non-i.i.d. extension. Finally, we discuss conditions for the system reliability to have both representations. [less ▲] Detailed reference viewed: 23 (7 ULg)A description of n-ary semigroups polynomial-derived from integral domains ; Mathonet, Pierre in Semigroup Forum (2011) We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Glazek and Gleichgewicht ... [more ▼] We provide a complete classification of the n-ary semigroup structures defined by polynomial functions over infinite commutative integral domains with identity, thus generalizing Glazek and Gleichgewicht's classification of ternary semigroups. [less ▲] Detailed reference viewed: 22 (6 ULg)Weighted Banzhaf power and interaction indexes through weighted approximations of games ; Mathonet, Pierre in European Journal of Operational Research (2011), 211 The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside ... [more ▼] The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes. [less ▲] Detailed reference viewed: 16 (2 ULg)Measuring the interactions among variables of functions over the unit hypercube ; Mathonet, Pierre in Journal of Mathematical Analysis & Applications (2011), 380 By considering a least squares approximation of a given square integrable function f: [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: [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 several 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. Finally, we discuss a few applications of the interaction index in aggregation function theory. [less ▲] Detailed reference viewed: 13 (2 ULg)Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations ; Mathonet, Pierre in Journal of Multivariate Analysis (2011), 102 The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression ... [more ▼] The concept of system signature was introduced by Samaniego for systems whose components have i.i.d. lifetimes. We consider its extension to the continuous dependent case and give an explicit expression for this extension as a difference of weighted means of the structure function values. We then derive a formula for the computation of the coefficients of these weighted means in the special case of independent continuous lifetimes. Finally, we interpret this extended concept of signature through a natural least squares approximation problem. [less ▲] Detailed reference viewed: 17 (1 ULg)Measuring the Interactions among Variables of Functions over the Unit hypercube ; Mathonet, Pierre in Modeling Decisions for Artificial Intelligence 7th International Conference, MDAI 2010 (2010) By considering a least squares approximation of a given square integrable function f:[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:[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. Finally, we discuss a few applications of the interaction index. [less ▲] Detailed reference viewed: 11 (0 ULg)Measuring the Influence of the kth Largest Variable on Functions over the Unit Hypercube ; Mathonet, Pierre in Modeling Decisions for Artificial Intelligence 7th International Conference, MDAI 2010 (2010) By considering a least squares approximation of a given square integrable function f: [0,1]^n\to R by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index ... [more ▼] By considering a least squares approximation of a given square integrable function f: [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: 3 (0 ULg)Approximations of Lovász extensions and their induced interaction index ; Mathonet, Pierre in Discrete Applied Mathematics (2008), 156(1), 11-24 The Lovasz 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 (f) over cap : [0, 1](n) -> R that ... [more ▼] The Lovasz 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 (f) over cap : [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 Lovasz extension (f) over cap by Lovasz 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 [Approximations of pseudo-Boolean functions; applications to game theory, Z. Oper. Res. 36(1) (1992) 3-21] and then solved explicitly by Grabisch et al. [Equivalent representations of set functions, Math. Oper. Res. 25(2) (2000) 157-178], giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of (f) over cap 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 [How to improve acts: an alternative representation of the importance of criteria in MCDM, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 9(2) (2001) 145-157]. (c) 2007 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 64 (10 ULg)On Comparison Meaningfulness of Aggregation Functions ; Mathonet, Pierre in Journal of Mathematical Psychology (2001), 45(2), 213-223 This paper will give a description of all continuous functions which are comparison meaningful in the sense of measurement theory. Detailed reference viewed: 16 (1 ULg)A characterization of the ordered weighted averaging functions based on the ordered bisymmetry property ; Mathonet, Pierre in IEEE Transactions on Fuzzy Systems (1999), 7(1), Detailed reference viewed: 13 (2 ULg)Characterization of some functions stable for positive linear transformations ; Mathonet, Pierre ; in Fuzzy Sets & Systems (1999), 102 This paper deals with a characterization of a class of aggregation operators. This class concerns operators which are symmetric, increasing, stable for the same positive linear transformations and present ... [more ▼] This paper deals with a characterization of a class of aggregation operators. This class concerns operators which are symmetric, increasing, stable for the same positive linear transformations and present a property close to the bisymmetry property: the ordered bisymmetry property. It is proved that the class investigated contains exactly the ordered weighted averaging operators (OWA) introduced by Yager in 1988. [less ▲] Detailed reference viewed: 11 (2 ULg) |
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