Generalization of one algorithm for constructing recurrent splines

Galina Tuluchenko, Gennadii Virchenko, Galyna Getun, Viacheslav Martynov, Mykola Tymofieiev

Abstract


We have analyzed two algorithms, close in composition, for constructing a smoothing spline, which imply a change only in the last link of the spline when new experimental data arrive. The main feature of the N. D. Dicoussar algorithm is the form of a polynomial representation in order to describe a link of the spline. It is shown that a given polynomial is one of the hierarchical form of the Hermitian polynomial.

We have proposed a modification to the D. A. Silaev algorithm for constructing a smoothing spline with different orders of smoothness: from zero to the second, aimed at enhancing the stability of this algorithm. To this end, we substantiated recommendations related to the form of polynomials representation, which describe the links of splines of the specified form. For this purpose, we estimated conditionality of matrices used in the algorithm. For the spline of zero-order smoothness, the most advisable is to apply a polynomial in the N. D. Dicoussar form, and for splines with higher orders of smoothness of joining the links, it is appropriate to use different forms of the Hermitian polynomials.

Based on computational examples, a possibility was demonstrated to generalize the D. A. Silaev algorithm to construct a spline with links of various lengths, which is determined by the rate of change in the examined parameter. That makes it possible to reduce the volume of information that contains a description of the spline itself, and to prevent such a widespread shortcoming of approximation when using polynomials as parasitic oscillations. It was shown as well that in the presence of significant measurement errors in experimental data there may occur a need to decrease the length of the spline's link (compared to that derived by the D. A. Silaev rule) in order to provide the spline with a property of robustness

Keywords


smoothing spline; a time series; algorithm stability; matrix conditionality; algorithm resource-intensity

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References


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GOST Style Citations


Wang Y. Smoothing Splines: Methods and Applications. New York, 2011. 384 p. doi: 10.1201/b10954 

Smoothing spline analysis of variance models: A new tool for the analysis of cyclic biomechanical data / Helwig N. E., Shorter K. A., Ma P., Hsiao-Wecksler E. T. // Journal of Biomechanics. 2016. Vol. 49, Issue 14. P. 3216–3222. doi: 10.1016/j.jbiomech.2016.07.035 

Liu Z., Guo W. Data Driven Adaptive Spline Smoothing. Statistica Sinica. 2010. Issue 20. P. 1143–1163.

Knot selection for least-squares and penalized splines / Spiriti S., Eubank R., Smith P. W., Young D. // Journal of Statistical Computation and Simulation. 2013. Vol. 83, Issue 6. P. 1020–1036. doi: 10.1080/00949655.2011.647317 

Yuan Y., Chen N., Zhou S. Adaptive B-spline knot selection using multi-resolution basis set // IIE Transactions. 2013. Vol. 45, Issue 12. P. 1263–1277. doi: 10.1080/0740817x.2012.726758 

Knot calculation for spline fitting via sparse optimization / Kang H., Chen F., Li Y., Deng J., Yang Z. // Computer-Aided Design. 2015. Vol. 58. P. 179–188. doi: 10.1016/j.cad.2014.08.022 

Brandt C., Seidel H.-P., Hildebrandt K. Optimal Spline Approximation via ℓ0-Minimization // Computer Graphics Forum. 2015. Vol. 34, Issue 2. P. 617–626. doi: 10.1111/cgf.12589 

Bureick J., Alkhatib H., Neumann I. Robust Spatial Approximation of Laser Scanner Point Clouds by Means of Free-form Curve Approaches in Deformation Analysis // Journal of Applied Geodesy. 2016. Vol. 10, Issue 1. doi: 10.1515/jag-2015-0020 

Dikusar N. D. Optimizaciya resheniya v zadachah kusochno-polinomial'noy approksimacii. Dubna, 2016. 14 p.

Silaev D. A. Polulokal'niye sglazhivayushchie splayny // Trudy seminara im. I. G. Petrovskogo. 2013. Issue 29. P. 443–454.

Silaev D. A. Polulokal'nye sglazhivayushchie S-splayny // Komp'yuternye issledovaniya i modelirovanie. 2010. Vol. 2, Issue 4. P. 349–357.

Polulokal'nye sglazhivayushchie splayny klassa S1 / Silaev D. A. et. al. // Trudy seminara imeni I. G. Petrovskogo. 2007. Issue 26. P. 347–367.

Liu X., Yang Q., Jin H. New Representations of the Group Inverse of 2×2 Block Matrices // Journal of Applied Mathematics. 2013. Vol. 2013. P. 1–10. doi: 10.1155/2013/247028 

Pinezhaninov F., Pinezhaninov P. Bazisnye funkcii dlya konechnyh elementov. URL: http://old.exponenta.ru/soft/mathemat/pinega/a1/a1.asp

Van Manh P. Hermite interpolation with symmetric polynomials // Numerical Algorithms. 2017. Vol. 76, Issue 3. P. 709–725. doi: 10.1007/s11075-017-0278-0 



DOI: https://doi.org/10.15587/1729-4061.2018.128312

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Copyright (c) 2018 Galina Tuluchenko, Gennadii Virchenko, Galyna Getun, Viacheslav Martynov, Mykola Tymofieiev

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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061