Skip to main content

Measuring the Interactions among Variables of Functions over the Unit Hypercube

  • Conference paper
Modeling Decisions for Artificial Intelligence (MDAI 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6408))

Abstract

By considering a least squares approximation of a given square integrable function \(f : [0,1]^n \rightarrow {I \kern -3pt \mathcal R}\) 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Aiken, L.S., West, S.G.: Multiple Regression: Testing and Interpreting Interactions. Sage Publications, Newbury Park (1991)

    Google Scholar 

  2. Banzhaf, J.F.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19, 317–343 (1965)

    Google Scholar 

  3. Ben-Or, M., Linial, N.: Collective coin flipping. In: Randomness and Computation, pp. 91–115. Academic Press, New York (1990); Earlier version: Collective coin flipping, robust voting games and minima of Banzhaf values. In: Proc. 26th IEEE Symposium on the Foundation of Computer Sciences, Portland, pp. 408–416 (1985)

    Google Scholar 

  4. Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y., Linial, N.: The influence of variables in product spaces. Isr. J. Math. 77(1-2), 55–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dubey, P., Shapley, L.S.: Mathematical properties of the Banzhaf power index. Math. Oper. Res. 4, 99–131 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fujimoto, K., Kojadinovic, I., Marichal, J.-L.: Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games Econom. Behav. 55(1), 72–99 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grabisch, M., Labreuche, C.: How to improve acts: An alternative representation of the importance of criteria in MCDM. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 9(2), 145–157 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation functions. Encyclopedia of Mathematics and its Applications, vol. 127. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  9. Grabisch, M., Marichal, J.-L., Roubens, M.: Equivalent representations of set functions. Math. Oper. Res. 25(2), 157–178 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grabisch, M., Roubens, M.: An axiomatic approach to the concept of interaction among players in cooperative games. Int. J. Game Theory 28(4), 547–565 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hammer, P., Holzman, R.: Approximations of pseudo-Boolean functions; applications to game theory. Z. Oper. Res. 36(1), 3–21 (1992)

    MathSciNet  MATH  Google Scholar 

  12. Hammer, P., Rudeanu, S.: Boolean methods in operations research and related areas. Springer, Heidelberg (1968)

    Book  MATH  Google Scholar 

  13. Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions. In: Proc. 29th Annual Symposium on Foundations of Computational Science, pp. 68–80. Computer Society Press (1988)

    Google Scholar 

  14. Marichal, J.-L.: The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making. Discrete Appl. Math. 107(1-3), 139–164 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Marichal, J.-L., Kojadinovic, I., Fujimoto, K.: Axiomatic characterizations of generalized values. Discrete Applied Mathematics 155(1), 26–43 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Marichal, J.-L., Mathonet, P.: Approximations of Lovász extensions and their induced interaction index. Discrete Appl. Math. 156(1), 11–24 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shapley, L.: A value for n-person games. In: Contributions to the Theory of Games II. Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton University Press, Princeton (1953)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Marichal, JL., Mathonet, P. (2010). Measuring the Interactions among Variables of Functions over the Unit Hypercube. In: Torra, V., Narukawa, Y., Daumas, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2010. Lecture Notes in Computer Science(), vol 6408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16292-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-16292-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16291-6

  • Online ISBN: 978-3-642-16292-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics