Abstract
Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.
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Whittaker, E. T.,On a New Method of Graduation, Proceedings of the Edinburgh Mathematical Society, Vol. 41, pp. 63–75, 1923.
Schoenberg, I. J.,Spline Functions and the Problem of Graduation, Proceedings of the National Academy of Sciences, Vol. 52, pp. 947–950, 1964.
Anselone, P. M., andLaurent, P. J.,A General Method for the Construction of Interpolating or Smoothing Spline Functions, Numerische Mathematik, Vol. 12, pp. 66–82, 1968.
Kimeldorf, G., andWahba, G.,Some Results on Tchebycheffian Spline Functions, Journal of Mathematical Analysis and Applications, Vol. 33, pp. 82–95, 1971.
Reinsch, C.,Smoothing by Spline Functions, II, Numerische Mathematik, Vol. 16, pp. 451–454, 1971.
Lyche, T., andSchumaker, L. L.,Computation of Smoothing and Interpolating Natural Splines Via Local Bases, SIAM Journal on Numerical Analysis, Vol. 10, pp. 1027–1038, 1973.
Munteanu, M. J.,Generalized Smoothing Spline Functions for Operators, SIAM Journal on Numerical Analysis, Vol. 10, pp. 28–34, 1973.
Defigueiredo, R. J. P., andCaprihan, A.,An Algorithm for the Construction of the Generalized Smoothing Spline, Proceedings of the Johns Hopkins Conference on Information Sciences and Systems, pp. 494–500, 1977.
Wahba, G., andWold, S.,A Completely Automatic French Curve: Fitting Spline Functions by Cross Validation, Communications in Statistics, Vol. 4, pp. 1–17, 1975.
Weinert, H. L., Byrd, R. H., andSidhu, G. S.,Estimation Techniques for Recursive Smoothing of Deterministic Data, Proceedings of the Johns Hopkins Conference on Information Sciences and Systems, pp. 54–57, Baltimore, Maryland, 1977.
Weinert, H. L., andSidhu, G. S.,A Stochastic Framework for Recursive Computation of Spline Functions: Part I, Interpolating Splines, IEEE Transactions on Information Theory, Vol. 24, pp. 45–50, 1978.
Weinert, H. L., andSidhu, G. S.,On Uniqueness Conditions for Optimal Curve Fitting, Journal of Optimization Theory and Applications, Vol. 23, pp. 211–216, 1977.
Aronszajn, N.,Theory of Reproducing Kernels, Transactions of the American Mathematical Society, Vol. 68, pp. 337–404. 1950.
Parzen, E.,An Approach to Time Series Analysis, Annals of Mathematical Statistics, Vol. 32, pp. 951–989, 1961.
Weinert, H. L.,Statistical Methods in Optimal Curve Fitting, Communications in Statistics, Vol. B7, pp. 417–435, 1978.
Weinert, H. L., Desai, U. B., andSidhu, G. S.,ARMA Splines, System Inverses, and Least-Squares Estimates, SIAM Journal on Control and Optimization, Vol. 17, pp. 525–536, 1979.
Sidhu, G. S., andWeinert, H. L.,Vector-Valued Lg-Splines, Journal of Mathematical Analysis and Applications, Vol. 70, pp. 505–529, 1979.
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Communicated by D. G. Luenberger
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Weinert, H.L., Byrd, R.H. & Sidhu, G.S. A stochastic framework for recursive computation of spline functions: Part II, smoothing splines. J Optim Theory Appl 30, 255–268 (1980). https://doi.org/10.1007/BF00934498
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DOI: https://doi.org/10.1007/BF00934498