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Revisiting fitting monotone polynomials to data

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  • Kevin Murray
  • Samuel Müller
  • Berwin Turlach

Abstract

We revisit Hawkins’ (Comput Stat 9(3):233–247, 1994 ) algorithm for fitting monotonic polynomials and discuss some practical issues that we encountered using this algorithm, for example when fitting high degree polynomials or situations with a sparse design matrix but multiple observations per $$x$$ -value. As an alternative, we describe a new approach to fitting monotone polynomials to data, based on different characterisations of monotone polynomials and using a Levenberg–Marquardt type algorithm. We consider different parameterisations, examine effective starting values for the non-linear algorithms, and discuss some limitations. We illustrate our methodology with examples of simulated and real world data. All algorithms discussed in this paper are available in the R Development Core Team (A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2011 ) package MonoPoly. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Kevin Murray & Samuel Müller & Berwin Turlach, 2013. "Revisiting fitting monotone polynomials to data," Computational Statistics, Springer, vol. 28(5), pages 1989-2005, October.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:5:p:1989-2005
    DOI: 10.1007/s00180-012-0390-5
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    References listed on IDEAS

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    1. J. O. Ramsay, 1998. "Estimating smooth monotone functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 365-375.
    2. S. McKay Curtis & Sujit K. Ghosh, 2011. "A variable selection approach to monotonic regression with Bernstein polynomials," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(5), pages 961-976, February.
    3. Dominik Heinzmann, 2008. "A filtered polynomial approach to density estimation," Computational Statistics, Springer, vol. 23(3), pages 343-360, July.
    4. Hazelton, Martin L. & Turlach, Berwin A., 2011. "Semiparametric regression with shape-constrained penalized splines," Computational Statistics & Data Analysis, Elsevier, vol. 55(10), pages 2871-2879, October.
    5. Friedman, Jerome H. & Hastie, Trevor & Tibshirani, Rob, 2010. "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(i01).
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    Cited by:

    1. Silvia Montagna & Vanessa Orani & Raffaele Argiento, 2021. "Bayesian isotonic logistic regression via constrained splines: an application to estimating the serve advantage in professional tennis," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(2), pages 573-604, June.
    2. Abdelaati Daouia & Hohsuk Noh & Byeong U. Park, 2016. "Data envelope fitting with constrained polynomial splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 3-30, January.
    3. Leah M. Feuerstahler, 2019. "Metric Transformations and the Filtered Monotonic Polynomial Item Response Model," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 105-123, March.
    4. Ng, Kenyon & Turlach, Berwin A. & Murray, Kevin, 2019. "A flexible sequential Monte Carlo algorithm for parametric constrained regression," Computational Statistics & Data Analysis, Elsevier, vol. 138(C), pages 13-26.

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    Keywords

    Monotone polynomial; Monotone regression;

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