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Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach Marichal, Jean-Luc ; 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 standard 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 standard least squares approximation of the function by a pseudo-Boolean function 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 certain 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: 99 (6 UL)On structure signatures and probability signatures of general decomposable systems (invited talk) Marichal, Jean-Luc ; Mathonet, Pierre ; Scientific Conference (2014, June 16) Detailed reference viewed: 86 (4 UL)On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes Marichal, Jean-Luc ; 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: 82 (14 UL)Computing system signatures through reliability functions Marichal, Jean-Luc ; 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: 100 (21 UL)On the cardinality index of fuzzy measures and the signatures of coherent systems Mathonet, Pierre ; Marichal, Jean-Luc in Mesiar, Radko; Pap, Endre; Klement, Erich Peter (Eds.) 34th Linz Seminar on Fuzzy Set Theory (LINZ 2013) - Non-Classical Measures and Integrals (2013) Detailed reference viewed: 48 (1 UL)A classification of bisymmetric polynomial functions over integral domains of characteristic zero Marichal, Jean-Luc ; Mathonet, Pierre in Aequationes Mathematicae (2012), 84(1-2), 125-136 We describe the class of n-variable polynomial functions that satisfy Aczél's bisymmetry property over an arbitrary integral domain of characteristic zero with identity. Detailed reference viewed: 84 (9 UL)Symmetric approximations of pseudo-Boolean functions with applications to influence indexes Marichal, Jean-Luc ; 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: 33 (4 UL)Influence and interaction indexes in cooperative games: a unified least squares approach Marichal, Jean-Luc ; Mathonet, Pierre Scientific Conference (2011, November 10) The classical Banzhaf power and interaction indexes used in cooperative game theory appear naturally as leading coefficients in the standard least squares approximation of the game under consideration by ... [more ▼] The classical Banzhaf power and interaction indexes used in cooperative game theory appear naturally as leading coefficients in the standard least squares approximation of the game under consideration by a set function of a specified degree. We observe that this still holds true if we consider approximations by set functions depending on specified variables. We show that the Banzhaf influence index can also be obtained from this new approximation problem. Considering certain weighted versions of this approximation, we also introduce a class of weighted Banzhaf influence indexes and analyze their most important properties. [less ▲] Detailed reference viewed: 28 (3 UL)On signature-based expressions of system reliability Marichal, Jean-Luc ; Mathonet, Pierre ; Waldhauser, Tamás 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: 64 (13 UL)A description of n-ary semigroups polynomial-derived from integral domains Marichal, Jean-Luc ; Mathonet, Pierre in Semigroup Forum (2011), 83(2), 241-249 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 the corresponding ternary semigroups. [less ▲] Detailed reference viewed: 58 (10 UL)Extensions of system signature and Barlow-Proschan importance index to dependent lifetimes Marichal, Jean-Luc ; Mathonet, Pierre Presentation (2011, September 07) Detailed reference viewed: 23 (2 UL)Signatures, decompositions of reliability, and approximation problems Marichal, Jean-Luc ; Mathonet, Pierre ; Waldhauser, Tamás Presentation (2011, September 07) Detailed reference viewed: 29 (0 UL)Measuring the interactions among variables of functions over the unit hypercube Marichal, Jean-Luc ; Mathonet, Pierre in Journal of Mathematical Analysis & Applications (2011), 380(1), 105-116 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: 43 (3 UL)Weighted Banzhaf power and interaction indexes through weighted approximations of games Marichal, Jean-Luc ; Mathonet, Pierre in European Journal of Operational Research (2011), 211(2), 352-358 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: 42 (3 UL)Classification of associative multivariate polynomial functions Marichal, Jean-Luc ; Mathonet, Pierre Scientific Conference (2011, June) Detailed reference viewed: 36 (2 UL)Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations Marichal, Jean-Luc ; Mathonet, Pierre in Journal of Multivariate Analysis (2011), 102(5), 931-936 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: 35 (2 UL)Indices de pouvoir et d'interaction en théorie des jeux coopératifs: une approche par moindres carrés Marichal, Jean-Luc ; Mathonet, Pierre Presentation (2011, January 18) Detailed reference viewed: 31 (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: 34 (0 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: 26 (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: 31 (0 UL) |
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