Least squares approximation by splines with free knots

Schwetlick, Hubert; Schütze, Torsten

doi:10.1007/bf01732610

Cited by 76 publications

(39 citation statements)

References 17 publications

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“…We have performed a thorough simulation study of the impact on the GeDS knot location and related MSE (mean square error), of different assumptions and choices made in constructing the GeDS estimate, namely, different levels of the signal to noise ratio (SNR from 2 to 7), sample sizes (N = 90, 150, 256, 512, 2048) GeDS on a series of simulated examples, used in many other studies on variable knot spline methods (cf. Schwetlick and Schütze 1995, Fan and Gijbels 1995, Donoho and Johnstone 1994and Luo and Wahba 1997. Due to volume limitations, we present here the results for one of the simulated test functions, given in Table 1, first considered by Schwetlick and Schütze (1995), and one real data example from materials science, due to Kimber et al (2009).…”

Section: Numerical Studymentioning

confidence: 99%

“…Schwetlick and Schütze 1995, Fan and Gijbels 1995, Donoho and Johnstone 1994and Luo and Wahba 1997. Due to volume limitations, we present here the results for one of the simulated test functions, given in Table 1, first considered by Schwetlick and Schütze (1995), and one real data example from materials science, due to Kimber et al (2009). The detailed results related to the other test functions are given in the online supplement to this paper, see Tables 1 and 2 therein.…”

Section: Numerical Studymentioning

confidence: 99%

“…In this section, we aim at illustrating the method and its properties on the following test example, which appears in Schwetlick and Schütze (1995). The proposed GeDS method has been implemented using Mathematica 9.0 and a standard PC (Intel core i7 CPU, 2.93 Ghz, 8GB RAM) has been used for all test examples.…”

Section: A Simulated Examplementioning

confidence: 99%

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Geometrically designed, variable knot regression splines

et al. 2015

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractA new method of Geometrically Designed least squares (LS) splines with variable knots, named GeDS, is proposed. It is based on the property that the spline regression function, viewed as a parametric curve, has a control polygon and, due to the shape preserving and convex hull properties, it closely follows the shape of this control polygon. The latter has vertices whose x-coordinates are certain knot averages and whose y-coordinates are the regression coefficients. Thus, manipulation of the position of the control polygon may be interpreted as estimation of the spline curve knots and coefficients. These geometric ideas are implemented in the two stages of the GeDS estimation method. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order (n > 2) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible and finally, the LS estimates of the regression coefficients of this curve are found. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline coefficients. Numerical examples are provided and further supplemental materials are available online.

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Section: Numerical Studymentioning

confidence: 99%

Section: Numerical Studymentioning

confidence: 99%

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Geometrically designed, variable knot regression splines

et al. 2015

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“…A classical way to extend least squares fitting methods to force them to produce smooth surfaces is to add a smoothing term in terms of a Sobolev seminorm for H 1 or H 2 to the functional (1), see, e.g., [10] for general splines, [35] for splines with free knots, [19] for hierarchical splines, [21] for splines on triangulations, or [16] in a wavelet reformulation of the spline problem. Following [28], it is more advantageous in view of the norm equivalence (4) to consider here a cost functional of the form…”

Section: Smoothing With Waveletsmentioning

confidence: 99%

Multilevel regularization of wavelet based fitting of scattered data ? some experiments

Castaño

Kunoth

2005

Numer Algor

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In [6], an adaptive method to approximate unorganized clouds of points by smooth surfaces based on wavelets has been described. The general fitting algorithm operates on a coarse-tofine basis. It selects on each refinement level in a first step a reduced number of wavelets which are appropriate to represent the features of the data set. In a second step, the fitting surface is constructed as the linear combination of the wavelets which minimizes the distance to the data in a least squares sense. This is followed by a thresholding procedure on the wavelet coefficients to discard those which are too small to contribute much to the surface representation.In this paper, we firstly generalize this strategy to a classically regularized least squares functional by adding a Sobolev norm, taking advantage of the capability of wavelets to characterize Sobolev spaces of even fractional order. After recalling the usual cross-validation technique to determine the involved smoothing parameters, some examples of fitting severely irregularly distributed data, synthetically produced and of geophysical origin, are presented. In order to reduce computational costs, we then introduce a multilevel generalized crossvalidation technique which goes beyond the Sobolev formulation and exploits the hierarchical setting based on wavelets. We illustrate the performance of the new strategy on some geophysical data.

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“…This algorithm only gave an optimal distribution for knots. Schwetlick and Schütze (1995) obtained the places of knots by using nonlinear optimization method when the number of knots was given. Also, Lindstrom (1999) defined a nonlinear optimization algorithm based on free knot spline modeling.…”

Section: Introductionmentioning

confidence: 99%

Variable selection with genetic algorithm and multivariate adaptive regression splines in the presence of multicollinearity

Kılınç

Aşikgil²,

Erar

et al. 2016

Int. j. adv. appl. sci

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In this paper, it is aimed to determine the true regressors explaining the dependent variable in multiple linear regression models and also to find the best model by using two different approaches in the presence of low, medium and high multicollinearity. These approaches compared in this study are genetic algorithm and multivariate adaptive regression splines. A comprehensive Monte Carlo experiment is performed in order to examine the performance of these approaches. This study exposes that nonparametric methods can be preferred for variable selection in order to obtain the best model when there is a multicollinearity problem in the small, medium or large data sets.

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Least squares approximation by splines with free knots

Cited by 76 publications

References 17 publications

Geometrically designed, variable knot regression splines

Geometrically designed, variable knot regression splines

Multilevel regularization of wavelet based fitting of scattered data ? some experiments

Variable selection with genetic algorithm and multivariate adaptive regression splines in the presence of multicollinearity

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