k-order additive discrete fuzzy measures and their representation
Fuzzy Sets and Systems - Special issue on fuzzy measures and integrals
Equivalent Representations of Set Functions
Mathematics of Operations Research
How to improve ACTS: an alternative representation of the importance of criteria in MCDM
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems - Special issue on aggregation operators
Aggregation of interacting criteria by means of the discrete Choquet integral
Aggregation operators
Measuring the interactions among variables of functions over the unit hypercube
MDAI'10 Proceedings of the 7th international conference on Modeling decisions for artificial intelligence
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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@^:[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@^ 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@^ 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].