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

Authors:
Adil Bagirov;Conny Clausen;Michael Kohler
Affiliations:
School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Australia 3353;Department of Mathematics, Universität des Saarlandes, Saarbrücken, Germany 66041;Department of Mathematics, Technische Universität Darmstadt, Darmstadt, Germany 64289
Venue:
Computational Optimization and Applications
Year:
2010

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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.