Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:20:30.775Z Has data issue: false hasContentIssue false
  • English
  • Français

On the equivalence of two stochastic approaches to spline smoothing

Published online by Cambridge University Press:  14 July 2016

Craig F. Ansley  and
Robert Kohn
Get access Rights & Permissions [Opens in a new window]

Abstract

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

Keywords

SIGNAL EXTRACTIONDIFFUSE PRIORSTOCHASTIC DIFFERENTIAL EQUATIONNON-LINEAR REGRESSIONKALMAN FILTER
Type
Part 7—Algorithms and Computations
Information
Journal of Applied Probability , Volume 23 , Issue A: Essays in Time Series and Allied Processes , 1986 , pp. 391 - 405
Copyright
Copyright © 1986 Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anselone, P. M. and Laurent, P. J. (1968) A general method for the construction of interpolating or smoothing spline functions. Numer. Math. 12, 66–82.Google Scholar
Ansley, C. F. and Kohn, R. (1983a) State space models with diffuse initial conditions I: Filtering and likelihood. Technical Report #13, Statistics Research Center, Graduate School of Business, University of Chicago.Google Scholar
Ansley, C. F. and Kohn, R. (1983b) State space models with diffuse initial conditions II: Smoothing and continuous time processes. Technical Report #15, Statistics Research Center, Graduate School of Business, University of Chicago.Google Scholar
Ansley, C. F. and Kohn, R. (1984) The equivalence of two stochastic approaches to spline smoothing. Technical Report #19, Statistics Research Center, Graduate School of Business, University of Chicago.Google Scholar
Coddington, E. A. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.Google Scholar
Craven, P. and Wahba, G. (1979) Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 317–403.Google Scholar
Kimeldorf, G. and Wahba, G. (1970a) A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Statist. 41, 495–502.Google Scholar
Kimeldorf, G. and Wahba, G. (1970b) Spline functions and stochastic processes. Sankhya A 32, 173–180.Google Scholar
Kimeldorf, G. and Wahba, G. (1971) Some results on Tchebycheffian spline functions. J. Multivariate Anal. Appl. 33, 82–95.Google Scholar
Kohn, R. and Ansley, C. F. (1983) On the smoothness properties of the best linear unbiassed estimate of a stochastic process observed with noise. Ann. Statist. 11, 1011–1017.Google Scholar
Kohn, R. and Ansley, C. F. (1984) A new algorithm for spline smoothing and interpolation based on smoothing a stochastic process. SIAM J. Scient. Statist. Comput. To appear.Google Scholar
Schoenberg, I. J. (1964) Spline functions and the problem of graduation. Proc. Nat. Acad. Sci. USA 52, 947–950.Google Scholar
Sidhu, G. S. and Weinert, H. L. (1979) Vector-valued Lg-splines I: Interpolating splines. J. Math. Anal. Appl. 70, 505–529.Google Scholar
Sidhu, G. S. and Weinert, H. L. (1984) Vector-valued Lg-splines II: Smoothing splines. J. Math. Anal. Appl. 101, 380–396.Google Scholar
Silverman, B. W. (1984) A fast and efficient cross-validation method for smoothing parameter choice in spline regression. J. Amer. Statist. Assoc. 79, 584–589.CrossRefGoogle Scholar
Wahba, G. (1978) Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. R. Statist. Soc. B 40, 364–372.Google Scholar
Wahba, G. (1983a) Bayesian confidence intervals for the cross validated smoothing spline. J. R. Statist. Soc. B 45, 133–150.Google Scholar
Wahba, G. (1983b) A comparison of GCV and GML for choosing the smoothing parameter in the general spline smoothing problem. Technical Report #712, Department of Statistics, University of Wisconsin, Madison.Google Scholar
Wecker, W. E. and Ansley, C. F. (1983) The signal extraction approach to nonlinear regression and spline smoothing. J. Amer. Statist. Assoc. 78, 81–89.Google Scholar
Weinert, H. L., Byrd, R. H. and Sidhu, G. S. (1980) A stochastic framework for recursive computation of spline functions: Part II, Smoothing splines. J. Optimization Theory Appl. 30, 255–268.CrossRefGoogle Scholar
Weinert, H. L. and Sidhu, G. S. (1979) A stochastic framework for recursive computation of spline functions: Part I, Interpolating splines. IEEE Trans. Information Theory IT-24, 45–50.Google Scholar