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Smoothing noisy data with spline functions

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Abstract

It is shown how to choose the smoothing parameter when a smoothing periodic spline of degree 2m−1 is used to reconstruct a smooth periodic curve from noisy ordinate data. The noise is assumed “white”, and the true curve is assumed to be in the Sobolev spaceW (2m)2 of periodic functions with absolutely continuousv-th derivative,v=0, 1, ..., 2m−1 and square integrable 2m-th derivative. The criteria is minimum expected square error, averaged over the data points. The dependency of the optimum smoothing parameter on the sample size, the noise variance, and the smoothness of the true curve is found explicitly.

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References

  1. Anderssen, R. S., Bloomfield, P.: A time series approach to numerical differentiation. Technometrics,16, 69–75 (1974)

    Google Scholar 

  2. Anderssen, R. S., Bloomfield, P.: Numerical differentiation procedures for nonexact data. Numer. Math.22, 157–182 (1974)

    Google Scholar 

  3. Cogburn, R., Davis, H. T.: Periodic Splines and Spectral Estimation. Ann. Statist.2, 1108–1126 (1974)

    Google Scholar 

  4. Golomb, M.: Approximation by periodic spline interpolants on uniform meshes. J. Approx. Thy.1, 26–65 (1968)

    Article  Google Scholar 

  5. Kershaw, D.: A note on the convergence of interpolatory cubic splines, Siam J. Numer. Anal.,8, 67–74 (1971)

    Article  Google Scholar 

  6. Kimeldorf, G., Wahba, G.: Some results on Tchebycheffian spline functions, J. Math. Anal. Appl.33, 82–95 (1971)

    Article  Google Scholar 

  7. Jolley, L. B. W.: Summation of Series. Second revised ed., Dover, New York 1961.

    Google Scholar 

  8. Reinsch, C. H.: Smoothing by spline functions. Numer Math.10, 177–183 (1967)

    Google Scholar 

  9. Reinsch, C. H.: Smoothing by Spline Functions. II. Numer. Math.16, 451–454 (1971)

    Google Scholar 

  10. Schoenberg, I. J.: Spline functions and the problem of graduation. Proc. Nat. Acad. Sci. (U.S.A.)52, 947–950 (1964)

    Google Scholar 

  11. Schultz, M.: Error bounds for polynomial spline interpolation. Math. Comp.24, 111, 507–515 (1970)

    Google Scholar 

  12. Snowden, M.: M.S. Thesis, in preparation

  13. Wahba, G.: Convergence properties of the method of regularization for noisy linear operator equations, Univ. of Wisconsin Mathematics Research Center TSR 1132 (1973)

  14. Wahba, G., Wold, S.: A completely automatic French curve: Fitting spline functions by cross-validation. Communications in Statistics4, 1–17 (1975)

    Google Scholar 

  15. Wahba, G., Wold, S.: Periodic splines for spectral density estimation: The use of cross validation for determining the degree of smoothing. Communications in Statistics4 (2), 125–141 (1975)

    Google Scholar 

  16. Wiencke, B.: “The roundoff: A Mechanical Analysis of a Skillfully Executed Gymnastic Stunt”, unpublished M. S. thesis, Women's Physical Education Department University of Wisconsin, Madison (1972)

    Google Scholar 

  17. Wold, S.: Spline functions in data analysis, Technometrics16, 1–11 (1974)

    Google Scholar 

  18. Wold, S.: Estimation of activation parameters from one kinetic experiment (Varytemp method). Error analysis and revised computer program. Acta. Chem. Scand.25, 336–339 (1971)

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Statistics, Stanford University, 94303, Stanford, Calif, USA

    Grace Wahba

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  1. Grace Wahba
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This work was supported by the U. S. Air Force Office of Scientific Research under Grant AF-AFOSR-2363-B.

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Wahba, G. Smoothing noisy data with spline functions. Numer. Math. 24, 383–393 (1975). https://doi.org/10.1007/BF01437407

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