Reference : Approximations of Lovász extensions and their induced interaction index
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Business & economic sciences : Quantitative methods in economics & management
http://hdl.handle.net/10993/2850
Approximations of Lovász extensions and their induced interaction index
English
Marichal, Jean-Luc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Mathonet, Pierre mailto [University of Liège, Belgium > Department of Mathematics]
1-Jan-2008
Discrete Applied Mathematics
Elsevier Science
156
1
11-24
Yes (verified by ORBilu)
International
0166-218X
Amsterdam
The Netherlands
[en] pseudo-Boolean function ; Lovász extension ; discrete Choquet integral ; least squares approximation ; interaction index
[en] 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.
University of Luxembourg - UL
Researchers ; Professionals ; Students
http://hdl.handle.net/10993/2850
10.1016/j.dam.2007.07.011
http://arxiv.org/abs/0706.3856

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