2008
DOI: 10.1080/10485250801948625
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Approximating data with weighted smoothing splines

Abstract: Given a data set (t i , y i ), i = 1, . . . , n with the t i ∈ [0, 1] non-parametric regression is concerned with the problem of specifying a suitable function f n : [0, 1] → R such that the data can be reasonably approximated by the points (t i , f n (t i )), i = 1, . . . , 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 … Show more

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Cited by 19 publications
(18 citation statements)
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“…S := D k · L p ) which coincides with the minimax rate over Sobolev ellipsoids in W k,p up to a log-factor. Special cases of this result have also appeared in (Davies and Meise, 2008) for k = p = 2, and in (Davies et al, 2009) for k = 2, p = ∞. In particular, adaptation of this type of estimators has not been provided so far, to the best of our knowledge.…”
Section: Mind Estimatormentioning
confidence: 99%
“…S := D k · L p ) which coincides with the minimax rate over Sobolev ellipsoids in W k,p up to a log-factor. Special cases of this result have also appeared in (Davies and Meise, 2008) for k = p = 2, and in (Davies et al, 2009) for k = 2, p = ∞. In particular, adaptation of this type of estimators has not been provided so far, to the best of our knowledge.…”
Section: Mind Estimatormentioning
confidence: 99%
“…Apart from the taut-string algorithm [1], condition (2)-or similar ones-has been used in different nonparametric procedures like weighted smoothing splines [5] and piecewise constant least squares estimates penalised by the number of jumps [6]. Dümbgen and Spokoiny [7] suggest a test statistic closely related to Equation (2) for testing hypotheses concerning the shape of a nonparametric regression function; their derivation involves empirical process theory and the modulus of continuity of the Brownian motion.…”
Section: Introductionmentioning
confidence: 99%
“…Some properties of Equation (2) are treated in the literature, often asymptotic ones, cf. [1,5,6,9,10]. An efficient algorithm for checking whether Equation (2) is fulfilled is given by Bernholt and Coworkers [11,12].…”
Section: Introductionmentioning
confidence: 99%
“…Typically, the multiresolution criterion is implemented with an iterative procedure (Davies and Kovac, 2001;Davies and Meise, 2008), however, quadratic programming allows a straightforward implementation that makes it possible to avoid specifying a smoothing parameter but still apply an appropriate amount of smoothing. As the multiresolution criterion (2) is a system of linear inequalities it can be directly incorporated into the quadratic pro- In the case of combined total variation penalties (Section 3.1), we provide two global smoothing parameters λ and λ , and have to choose the ratio λ/λ , specifying how much we favour a smooth estimate over controlling local extremes.…”
Section: Multiresolutionmentioning
confidence: 99%