Skip to main content

Cross-validation techniques for smoothing spline functions in one or two dimensions

Part of the Lecture Notes in Mathematics book series (LNM,volume 757)

Abstract

This paper aims to present some results on the asymptotic behaviour of a matrix associated with certain types of spline functions and shows how these results can be used to obtain a fast algorithm for choosing the smoothing parameter in the smoothing of noisy data by splines.

First, we give a general theorem on the behaviour of the eigenvalues associated with a spline function. The spline functions considered here are those defined by a variational formulation (for general theorems on these splines see [1], [11]).

Second, we use the preceding results to obtain an O(n) — algorithm to calculate the optimal smoothing polynomial spline using the generalized Cross-Validation Technique, in the case of equally spaced data. We obtain an O(n2) algorithm to perform this calculation in the case of non equally spaced data. Some very good numerical results are presented and a table of run times is given. We then introduce a method which calculates the solution (smoothed data) by a piecewise smoothing and correction technique. This method allows us to treat a very important amount of data.

We show too, how calculations can be made with two dimensional data (with arbitrary data points) and generalize the piecewise calculations to this case. We also present a set of numerical results and a table of run times.

Keywords

  • Null Space
  • Spline Function
  • Thin Plate Spline
  • Polynomial Spline
  • Preceding Subsection

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographie

  1. ATTEIA M. "Théorie et Applications de Fonctions Spline en Analyse Numérique" Thèse. Grenoble (1966)

    Google Scholar 

  2. BABUSKA I & AZIZ A.K. "Foundations of the Finite Element Method with applications to Partial Differential Equations" Edited by A.K. AZIZ Academic Press, New York (1972)

    Google Scholar 

  3. CHATELIN F. "Théorie de l’Approximation des Opérateurs Linéaires. Application au Calcul des Valeurs Propres d’Operateurs Differentiels et Integraux". Cours de DEA d’Analyse Numérique. Université Scientifique et Médicale de Grenoble (1966)

    Google Scholar 

  4. COURANT R. & HILBERT D. "Methods of Mathematical Physics" Vol I. Interscience Publischers, New York (1953)

    MATH  Google Scholar 

  5. CRAVEN P. & WAHBA G. "Smoothing Noisy Data with Spline Functions. Estimating the Correct Degree of Smoothing by the Method of Generalized Cross-Validation" Numerische Mathematik 31, 317–403 (1979)

    MathSciNet  MATH  Google Scholar 

  6. DUCHON J. "Fonctions Spline à Energie Invariante par Rotation" Rapport de Recherche no 27 U.S.M.G. Grenoble (1976)

    Google Scholar 

  7. DUCHON J. "Interpolation des Fonctions de Deux Variables par des Fonctions Spline du Type Plaque Mince" R.A.I.R.O. Analyse Numérique. Vol 10-no 12 (1976) pp 5–12

    MathSciNet  Google Scholar 

  8. FIX G. "Effects of Quadrature Errors in Finite Element Approximation of Steady State, Eigenvalue and Parabolic Problems" in "Foundations of the Finite Element Method with Applications to Partial Differential Equations" Edited by AZIZ A.K. Academic Press, New York (1972)

    Google Scholar 

  9. GOLUB G., HEATH M. & WAHBA G. "Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter" Technical Report no 491 May 1977 University of Winsconsin Madison.

    Google Scholar 

  10. KREISS H.O. "Difference Approximations for Boundary and Eigenvalue Problems for Ordinary Differential Equations" Mathematics of Computations, Vol 26, 1972, pp 605–624

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. LAURENT P.J. "Approximation et Optimisation" Hermman, Paris, 1972

    MATH  Google Scholar 

  12. NECAS J. "Les Méthodes Directes en Théorie des Equations Elliptiques" Masson, Paris, 1967

    Google Scholar 

  13. PAIHUA L. "Quelques Méthodes Numériques pour les Fonctions Spline à une et deux Variables" Thèse Grenoble. Mai 1978

    Google Scholar 

  14. PAIHUA L. & UTRERAS F. "Un Ensemble de Programmes pour l’Interpolation de Fonctions, par des Fonctions Spline du Type Plaque Mince" Rapport de Recherche no 140. Octobre 1978. IRMA Grenoble.

    Google Scholar 

  15. SCHWARTZ L. "Théorie des Distributions" Hermman, Paris, 1966.

    MATH  Google Scholar 

  16. SCHWARZ H.R. "Tridiagonalization of a Symmetric Band Matrix" in "Linear Algebra" Edited by J.H. WILKINSON et C. REINSCH Springer-Verlag. Berlin 1971

    Google Scholar 

  17. THOMMAN J. "Determination et Construction de Fonctions Spline à Deux Variables Définies sur un Domaine Rectangulaire ou Circulaire" Thèse Lille 1970.

    Google Scholar 

  18. UTRERAS F. "Sur le Choix du Paramètre d’Ajustement dans le Lissage par Fonctions Spline" Séminaire d’Analyse Numérique no 296 Grenoble Mars 1978.

    Google Scholar 

  19. UTRERASF. "Sur le Choix du Paramètre d’Ajustement dans le Lissage par Fonctions Spline" Exposé au Colloque National d’Analyse Numérique de France. Giens, le 21 Mai 1978 soumis pour publication.

    Google Scholar 

  20. UTRERAS F. "Utilisation des Programmes de Calcul du Paramètre d’Ajustement dans le Lissage par Fonctions Spline" Rapport de Recherche no 121 — Grenoble Mai 1978.

    Google Scholar 

  21. UTRERAS F. "Quelques Résultats d’Optimalité pour la Méthode de Validation Croisée" Séminaire d’Analyse Numérique no 301 Grenoble Novembre 1978

    Google Scholar 

  22. UTRERAS F. "Utilisation de la Méthode de Validation Croisée pour le Lissage par Fonctions Spline à une ou deux Variables" Doctoral Disertation. Université Scientifique et Médicale de Grenoble. Grenoble Mai 1979.

    Google Scholar 

  23. WAHBA G. "Practical Approximate Solutions to Linear Operator Equations when the Data are Noisy" SIAM Journal on Numerical Analysis — Vol 14 — no 4 Sept. 1977.

    Google Scholar 

  24. WAHBA G. "Smoothing Noisy Data with Spline Functions" Numerische Mathematik 24 (1975) pp 383–393

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. WAHBA G. "Optimal Smoothing of Density Estimates" in Classification and Clustering, J. VAN RYZIN ed., 423–458, Academic Press (1977)

    Google Scholar 

  26. WAHBA G. & WOLD S. "A Completely Automatic French Curve: Fitting Spline Functions by Cross-Validation" Comm. Statistics 4, 1–7, 1975.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1979 Springer-Verlag

About this paper

Cite this paper

Utreras, F. (1979). Cross-validation techniques for smoothing spline functions in one or two dimensions. In: Gasser, T., Rosenblatt, M. (eds) Smoothing Techniques for Curve Estimation. Lecture Notes in Mathematics, vol 757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098498

Download citation

  • DOI: https://doi.org/10.1007/BFb0098498

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09706-8

  • Online ISBN: 978-3-540-38475-5

  • eBook Packages: Springer Book Archive