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/10993/1461
Measuring the interactions among variables of functions over the unit hypercube
English
Marichal, Jean-Luc mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Mathonet, Pierre mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
1-Aug-2011
Journal of Mathematical Analysis & Applications
Elsevier
380
1
105-116
Yes
International
0022-247X
[en] interaction index ; multilinear polynomial ; least squares approximation ; difference operator ; aggregation function ; cooperative game ; fuzzy game
[en] 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.
Mathematics Research Unit
University of Luxembourg - UL
Researchers ; Professionals ; Students
http://hdl.handle.net/10993/1461
10.1016/j.jmaa.2011.02.040
http://arxiv.org/abs/0912.1547

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