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Measuring the interactions among variables of functions over the unit hypercube
Marichal, Jean-Luc; Mathonet, Pierre
2010In Torra, Vicenc; Narukawa, Yasuo; Daumas, Marc (Eds.) Modeling Decisions for Artificial Intelligence: Proceedings 7th International Conference, MDAI 2010, Perpignan, France, October 27-29, 2010
Peer reviewed
 

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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.
Disciplines :
Quantitative methods in economics & management
Mathematics
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
Language :
English
Title :
Measuring the interactions among variables of functions over the unit hypercube
Publication date :
19 October 2010
Event name :
7th Int. Conf. on Modeling Decisions for Artificial Intelligence (MDAI 2010)
Event organizer :
Vicenc Torra (IIIA-CSIC, South Catalonia, Spain)
Yasuo Narukawa (Toho Gakuen, Japan)
Event place :
Perpignan, France
Event date :
from 27-10-2010 to 29-10-2010
Audience :
International
Main work title :
Modeling Decisions for Artificial Intelligence: Proceedings 7th International Conference, MDAI 2010, Perpignan, France, October 27-29, 2010
Editor :
Torra, Vicenc
Narukawa, Yasuo
Daumas, Marc
Publisher :
Springer-Verlag, Berlin, Germany
ISBN/EAN :
978-3-642-16291-6
Collection name :
Lecture Notes in Artificial Intelligence Vol. 6408
Pages :
19-30
Peer reviewed :
Peer reviewed
Name of the research project :
F1R-MTH-PUL-09MRDO > MRDO > 01/01/2009 - 31/12/2011 > MARICHAL Jean-Luc
Funders :
University of Luxembourg - UL
Available on ORBilu :
since 29 September 2013

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