Article (Scientific journals)
Measuring the interactions among variables of functions over the unit hypercube
Marichal, Jean-Luc; Mathonet, Pierre
2011In Journal of Mathematical Analysis and Applications, 380, p. 105-116
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Keywords :
Interaction index; Multilinear Polynomial; Least Squares Approximation; Difference Operator; Aggregation Function; Cooperative fuzzy Game
Abstract :
[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.
Disciplines :
Mathematics
Author, co-author :
Marichal, Jean-Luc;  University of Luxembourg > FSTC > Mathematics Research Unit
Mathonet, Pierre ;  University of Luxembourg > FSTC > Mathematics Research Unit
Language :
English
Title :
Measuring the interactions among variables of functions over the unit hypercube
Publication date :
2011
Journal title :
Journal of Mathematical Analysis and Applications
ISSN :
0022-247X
eISSN :
1096-0813
Publisher :
Academic Press, San Diego, United States - California
Volume :
380
Pages :
105-116
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBi :
since 20 May 2011

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