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Approximating data with weighted smoothing splines

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  • P.L. Davies
  • M. Meise

Abstract

Given a data set (ti, yi), i=1, ˙s, n with ti∈[0, 1] non-parametric regression is concerned with the problem of specifying a suitable function fn:[0, 1]→ℝ such that the data can be reasonably approximated by the points (ti, fn(ti)), i=1, ˙s, n. If a data set exhibits large variations in local behaviour, for example large peaks as in spectroscopy data, then the method must be able to adapt to the local changes in smoothness. Whilst many methods are able to accomplish this, they are less successful at adapting derivatives. In this paper we showed how the goal of local adaptivity of the function and its first and second derivatives can be attained in a simple manner using weighted smoothing splines. A residual-based concept of approximation is used which forces local adaptivity of the regression function together with a global regularization which makes the function as smooth as possible subject to the approximation constraints.

Suggested Citation

  • P.L. Davies & M. Meise, 2008. "Approximating data with weighted smoothing splines," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(3), pages 207-228.
    • Handle: RePEc:taf:gnstxx:v:20:y:2008:i:3:p:207-228
    DOI: 10.1080/10485250801948625
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    References listed on IDEAS

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    1. Sally A. Wood, 2002. "Bayesian mixture of splines for spatially adaptive nonparametric regression," Biometrika, Biometrika Trust, vol. 89(3), pages 513-528, August.
    2. Alexandre Pintore & Paul Speckman & Chris C. Holmes, 2006. "Spatially adaptive smoothing splines," Biometrika, Biometrika Trust, vol. 93(1), pages 113-125, March.
    3. D. G. T. Denison & B. K. Mallick & A. F. M. Smith, 1998. "Automatic Bayesian curve fitting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 333-350.
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    Cited by:

    1. Hotz, Thomas & Marnitz, Philipp & Stichtenoth, Rahel & Davies, Laurie & Kabluchko, Zakhar & Munk, Axel, 2012. "Locally adaptive image denoising by a statistical multiresolution criterion," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 543-558.
    2. Thoralf Mildenberger, 2008. "A geometric interpretation of the multiresolution criterion in nonparametric regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(7), pages 599-609.

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