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An algorithm for the estimation of a regression function by continuous piecewise linear functions

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Abstract

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L 2 risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.

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References

  1. Bagirov, A.M.: Minimization methods for one class of nonsmooth functions and calculation of semi-equilibrium prices. In: Eberhard, A., et al. (eds.) Progress in Optimization: Contribution from Australia, pp. 147–175. Kluwer Academic, Dordrecht (1999)

    Google Scholar 

  2. Bagirov, A.M., Ugon, J.: Piecewise partially separable functions and a derivative-free method for large-scale nonsmooth optimization. J. Glob. Optim. 35, 163–195 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bagirov, A.M., Ghosh, M., Webb, D.: A derivative-free method for linearly constrained nonsmooth optimization. J. Ind. Manag. Optim. 2(3), 319–338 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Bagirov, A.M., Karasozen, B., Sezer, M.: Discrete gradient method: a derivative free method for nonsmooth optimization. J. Optim. Theory Appl. (2008). DOI: 10.1007/s10957-007-9335-5

    MathSciNet  Google Scholar 

  5. Bartels, S.G., Kuntz, L., Sholtes, S.: Continuous selections of linear functions and nonsmooth critical point theory. Nonlinear Anal. TMA 24, 385–407 (1995)

    Article  MATH  Google Scholar 

  6. Breiman, L., Friedman, J.H., Olshen, R.H., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont (1984)

    MATH  Google Scholar 

  7. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)

    Article  MATH  Google Scholar 

  8. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  9. Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Peter Lang, Frankfurt am Main (1995)

    MATH  Google Scholar 

  10. Friedman, J.H.: Multivariate adaptive regression splines (with discussion). Ann. Stat. 19, 1–141 (1991)

    Article  MATH  Google Scholar 

  11. Gorokhovik, V.V., Zorko, O.I., Birkhoff, G.: Piecewise affine functions and polyhedral sets. Optimization 31(3), 209–221 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Györfi, L., Kohler, M., Krzyżak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. Springer, Heldelberg (2002)

    MATH  Google Scholar 

  13. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, New York (2001)

    MATH  Google Scholar 

  14. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)

    Article  MathSciNet  Google Scholar 

  15. The R Project for Statistical Computing. Available on: www.r-project.org

  16. Wolfe, P.H.: Finding the nearest point in a polytope. Math. Program. 11, 128–149 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Information Technology and Mathematical Sciences, University of Ballarat, P.O. Box 663, Ballarat, Victoria, 3353, Australia

    Adil Bagirov

  2. Department of Mathematics, Universität des Saarlandes, Postfach 151150, 66041, Saarbrücken, Germany

    Conny Clausen

  3. Department of Mathematics, Technische Universität Darmstadt, Schloßgartenstr. 7, 64289, Darmstadt, Germany

    Michael Kohler

Authors
  1. Adil Bagirov
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  2. Conny Clausen
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  3. Michael Kohler
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Correspondence to Adil Bagirov.

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Bagirov, A., Clausen, C. & Kohler, M. An algorithm for the estimation of a regression function by continuous piecewise linear functions. Comput Optim Appl 45, 159–179 (2010). https://doi.org/10.1007/s10589-008-9174-9

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  • DOI: https://doi.org/10.1007/s10589-008-9174-9

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