Reference : Measuring the interactions among variables of functions over the unit hypercube
 Document type : Scientific journals : Article Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/1461
 Title : Measuring the interactions among variables of functions over the unit hypercube Language : English Author, co-author : Marichal, Jean-Luc [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Mathonet, Pierre [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Publication date : 1-Aug-2011 Journal title : Journal of Mathematical Analysis & Applications Publisher : Elsevier Volume : 380 Issue/season : 1 Pages : 105-116 Peer reviewed : Yes Audience : International ISSN : 0022-247X Keywords : [en] interaction index ; multilinear polynomial ; least squares approximation ; difference operator ; aggregation function ; cooperative game ; fuzzy game Abstract : [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. Research centres : Mathematics Research Unit Funders : University of Luxembourg - UL Target : Researchers ; Professionals ; Students Permalink : http://hdl.handle.net/10993/1461 DOI : 10.1016/j.jmaa.2011.02.040 Other URL : http://arxiv.org/abs/0912.1547

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