Reference : Measuring the interactions among variables of functions over the unit hypercube
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/2268/91468
Measuring the interactions among variables of functions over the unit hypercube
English
Marichal, Jean-Luc mailto [University of Luxembourg > FSTC > Mathematics Research Unit > >]
Mathonet, Pierre mailto [University of Luxembourg > FSTC > Mathematics Research Unit > >]
2011
Journal of Mathematical Analysis & Applications
Academic Press
380
105-116
Yes (verified by ORBi)
International
0022-247X
San Diego
CA
[en] Interaction index ; Multilinear Polynomial ; Least Squares Approximation ; Difference Operator ; Aggregation Function ; Cooperative fuzzy Game
[en] 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.
Researchers
http://hdl.handle.net/2268/91468
10.1016/j.jmaa.2011.02.040

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