Algorithm 642: A fast procedure for calculating minimum cross-validation cubic smoothing splines

ABSTRACT

The procedure CUBGCV is an implementation of a recently developed algorithm for fast O(n) calculation of a cubic smoothing spline fitted to n noisy data points, with the degree of smoothing chosen to minimize the expected mean square error at the data points when the variance of the error associated with the data is known, or, to minimize the generalized cross validation (GCV) when the variance of the error associated with the data is unknown. The data may be unequally spaced and nonuniformly weighted. The algorithm exploits the banded structure of the matrices associated with the cubic smoothing spline problem. Bayesian point error estimates are also calculated in O(n) operations.

REFERENCES

• 1: AMERICAN NATIONAL STANDARDS INSTITUTE. American National Standard programming language FORTRAN. ANSI X3.9-1978, American National Standards Institute, New York, 1978,
• 2: CRAVEN, e., AND WAHBA, G. Smoothing noisy data with spline functions. Numer. Math. 31 (1979), 377-403.
• 3: EUBANK, R.L. The hat maLrix for smoothing splines. Star. Probab. Lett. 2 (1984), 9-14.
• 4: HUTCHINSON, M. F., AND DE HOOG, F.R. Smoothing noisy data with spline functions. Numer. Math. 4 7 (1985), 99-106.
• 5: IMSL. IMSL Library Reference Manual. 9th ed., Houston, 1982.
• 6: REINSCH, C.H. Smoothing by spline functions. Numer. Math I0 (1967), 177-183.
• 7: SILVERMAN, n.W. Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J. R. Stat. Soc. B 47 (1985), 1-52.
• 8: UTRERAS, F. Sur le choix de parametre d'ajustement dans le lissage par fonctions spline. Numer. Math. 34 (1980), 15-28.
• 9: WAHBA, G. Ill-posed problems: Numerical and statistical methods for mildly, moderately, and severely ill-posed problems with noisy data. Univ. of Wisconsin-Madison Statistics Dept. Tech. Rep. 595, 1980. (To appear in Proceedings of International Conference on Ill-Posed Problems, M. Z. Nashed, Ed.)
• 10: WAHBA, G. Bayesian "confidence intervals" for the cross-validated smoothing spline. J. R. Star. Soc. B 45 (1983), 133-150.

INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Smoothing; Spline and piecewise polynomial interpolation
      G.1.2 Approximation
          Subjects: Spline and piecewise polynomial approximation
  G.3 PROBABILITY AND STATISTICS
      Subjects: Statistical software

General Terms:
Algorithms

Keywords:
Bayesian error estimates, banded matrices, cubic smoothing spline, generalized cross validation, influence matrix