zbMATH — the first resource for mathematics

Smoothing noisy data with spline functions. (English) Zbl 0578.65009
The authors present a procedure with a polynomial spline of degree 2m-1 for smoothing noisy data values observed at n distinct not necessarily equally spaced finite intervals. The procedure which minimizes the trace of the influence matrix associated with the smoothing spline requires just order \(m^ 2n\) operations, including the generalized cross validation, and order mn storage locations. Other methods require order \(n^ 3\) operations, therefore the described procedure is a significant improvement.
Reviewer: M.Nagel

65D10 Numerical smoothing, curve fitting
65D07 Numerical computation using splines
Full Text: DOI EuDML
[1] Craven, P., Wahba, G.: Smoothing noisy data with spline functions. Numer. Math.31, 377-403 (1979) · Zbl 0377.65007
[2] Curry, H.B., Schoenberg, I.J.: On polya frequency functions IV: The fundamental spline functions and their limits. J. Anal. Math.17, 71-107 (1966) · Zbl 0146.08404
[3] de Boor, C.: A Practical Guide to Splines. Appl. Math. Sci. vol. 27. New York: Springer 1978 · Zbl 0406.41003
[4] Dongarra, J.J., Moler, C.B., Bunch, J.R., Stewart, G.W.: Linpack User’s Guide. Philadelphia: Society for Industrial and Applied Mathematics 1979 · Zbl 0476.68025
[5] Eld?n, L.: An algorithm for the regularization of ill-conditioned banded least squares problems. SIAM J. Sci. Stat. Comput.5, 237-254 (1984) · Zbl 0546.65014
[6] Eld?n, L.: A note on the computation of the generalized cross-validation function for illconditioned least squares problems. BIT24, 467-472 (1984) · Zbl 0554.65029
[7] Erisman, A.M., Tinney, W.F.: On computing certain elements of the inverse of a sparse matrix. Commun. ACM18, 177-179 (1975) · Zbl 0296.65012
[8] Golub, G.H., Plemmons, R.J.: Large-scale geodetic least-squares adjustment by dissection and orthogonal decomposition. Lin. Alg. Appl.34, 3-27 (1980) · Zbl 0468.65012
[9] Haley, S.B.: Solution of band matrix equations by projection-recurrence. Lin. Alg. Appl.32, 33-48 (1980) · Zbl 0437.65023
[10] IMSL: Library Reference Manual, edition 9. Houston: IMSL 1982
[11] Reinsch, C.H.: Smoothing by spline functions. Numer. Math.10, 177-183 (1967) · Zbl 0161.36203
[12] Reinsch, C.H.: Smoothing by spline functions, II. Numer. Math.16, 451-454 (1971) · Zbl 1248.65020
[13] Schoenberg, I.J.: Spline functions and the problem of graduation. Proc. Natl. Acad. Sci. USA52, 947-950 (1964) · Zbl 0147.32102
[14] Takahashi, K., Fagan, J., Chin, M.-S.: Formation of a sparse bus impedance matrix and its application to short circuit study. Power Industry Computer Applications Conf. Proc. Minneapolis, Minn.8, 63-69 (June 4-6, 1973)
[15] Utreras, F.: Sur le choix de parametre d’adjustement dans le lissage par fonctions spline. Numer. Math.34, 15-28 (1980) · Zbl 0414.65007
[16] Utreras, F.: Optimal smoothing of noisy data using spline functions. SIAM J. Sci. Stat. Comput.2, 349-362 (1981) · Zbl 0476.65007
[17] Utreras, F.: Natural spline functions, their associated eigenvalue problem. Numer. Math.42, 107-117 (1983) · Zbl 0522.41011
[18] Wahba, G.: Smoothing noisy data with spline functions. Numer. Math.24, 383-392 (1975) · Zbl 0299.65008
[19] Wahba, G.: Bayesian ?confidence intervals? for the cross-validated smoothing spline. J.R. Stat. Soc., Ser. B45, 133-150 (1983) · Zbl 0538.65006
[20] Wecker, W.E., Ansley, C.F.: The signal extraction approach to non linear regression and spline smoothing. J. Am. Stat. Assoc.78, 81-89 (1983) · Zbl 0536.62071
[21] Weinert, H.L., Byrd, R.H., Sidhu, G.S.: A stochastic framework for recursive computation of spline functions: Part II, smoothing splines. J. Optimization Theory Appl.30, 255-268 (1980) · Zbl 0397.65008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.