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Information Geometric Perspective of Modal Linear Regression

  • Keishi Sando
  • Shotaro Akaho
  • Noboru Murata
  • Hideitsu Hino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11303)

Abstract

Modal linear regression (MLR) is a standard method for modeling the conditional mode of a response variable using a linear combination of explanatory variables. It is effective when dealing with response variables with an asymmetric, multi-modal distribution. Because of the nonparametric nature of MLR, it is difficult to construct a statistical model manifold in the sense of information geometry. In this work, a model manifold is constructed using observations instead of explicit parametric models. We also propose a method for constructing a data manifold based on an empirical distribution. The em algorithm, which is a geometric formulation of the EM algorithm, of MLR is shown to be equivalent to the conventional EM algorithm of MLR.

Keywords

Modal linear regression Information geometry EM algorithm 

References

  1. 1.
    Amari, S.: Information geometry of the EM and \(em\) algorithms for neural networks. Neural Netw. 8(9), 1379–1408 (1995)CrossRefGoogle Scholar
  2. 2.
    Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society (2000)Google Scholar
  3. 3.
    Baldauf, M., Silva, J.S.: On the use of robust regression in econometrics. Econ. Lett. 114(1), 124–127 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. Ser. B, 1–38 (1977)Google Scholar
  5. 5.
    Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics - The Approach Based on Influence Functions. Wiley (1986)Google Scholar
  6. 6.
    Huber, P.J., Ronchetti, E.M.: Robust Statistics. Wiley (2011)Google Scholar
  7. 7.
    Kemp, G.C., Silva, J.S.: Regression towards the mode. J. Econometrics 170(1), 92–101 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lee, M.J.: Mode regression. J. Econometrics 42(3), 337–349 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, J., Ray, S., Lindsay, B.G.: A nonparametric statistical approach to clustering via mode identification. J. Mach. Learn. Res. 8, 1687–1723 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Murata, N., Takenouchi, T., Kanamori, T., Eguchi, S.: Information geometry of u-boost and Bregman divergence. Neural Comput. 16(7), 1437–1481 (2004)CrossRefGoogle Scholar
  11. 11.
    Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statist. 23(5), 1543–1561 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Takano, K., Hino, H., Akaho, S., Murata, N.: Nonparametric e-mixture estimation. Neural Comput. 28(12), 2687–2725 (2016)CrossRefGoogle Scholar
  13. 13.
    Yao, W., Li, L.: A new regression model: modal linear regression. Scand. J. Stat. 41(3), 656–671 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.The Institute of Statistical MathematicsTachikawaJapan
  2. 2.University of TsukubaTsukubaJapan
  3. 3.National Institute of Advanced Industrial Science and TechnologyTsukubaJapan
  4. 4.Waseda UniversityShinjukuJapan

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