Regression Analysis Using a Blending Type Spline Construction

  • Tatiana KravetcEmail author
  • Børre Bang
  • Rune Dalmo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10521)


Regression analysis allows us to track the dynamics of change in measured data and to investigate their properties. A sufficiently good model allows us to predict the behavior of dependent variables with higher accuracy, and to propose a more precise data generation hypothesis.

By using polynomial approximation for big data sets with complex dependencies we get piecewise smooth functions. One way to obtain a smooth spline representation of an entire data set is to use local curves and to blend them using smooth basis functions. This construction allows the computation of derivatives at any point on the spline. Properties such as tangent, velocity, acceleration, curvature and torsion can be computed, which gives us the opportunity to exploit these data in the subsequent analysis.

We can adjust the accuracy of the approximation on the different segments of the data set by choosing a suitable knot vector. This article describes a new method for determining the number and location of the knot-points, based on changes in the Frenet frame.

We present a method of implementation using generalized expo-rational B-splines (GERBS) for regression problems (in two and three variables) and we evaluate the accuracy of the model using comparison of the residuals.


  1. 1.
    Bernstein, S.: Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités, Communications de la Société Mathématique de Kharkow, 2-ée série, vol. 13(1), pp. 1–2 (1912)Google Scholar
  2. 2.
    Bézier, P.: Numerical Control: Mathematics and Applications. Wiley series in Computing. Wiley, London, New York (1972). English language editionzbMATHGoogle Scholar
  3. 3.
    Bittner, K., Brachtendorf, H.G.: Fast algorithms for adaptive free-knot spline approximation using non-uniform biorthogonal spline wavelets. SIAM J. Sci. Comput. 37(2), 283–304 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    de Boor, C.: A Practical Guide to Splines. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bratlie, J., Dalmo, R., Zanaty, P.: Fitting of discrete data with GERBS. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2013. LNCS, vol. 8353, pp. 577–584. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43880-0_66 Google Scholar
  6. 6.
    Brezak, M., Petrovic, I.: Real-time approximation of clothoids with bounded error for path planning applications. IEEE Trans. Robot. 30, 507–515 (2014)CrossRefGoogle Scholar
  7. 7.
    Clarenz, U., Rumpf, M., Telea, A.: Robust feature detection and local classification for surfaces based on moment analysis. IEEE Trans. Vis. Comput. Graph. 10, 516–524 (2004)CrossRefGoogle Scholar
  8. 8.
    Dalmo, R.: Expo-Rational B-Splines in geometric modeling, methods for computer aided geometric design. Ph.D. thesis, University of Oslo (2016)Google Scholar
  9. 9.
    Dechevsky, L.T., Lakså, A., Bang, B.: Expo-rational B-splines. Int. J. Pure Appl. Math. 27(3), 319–367 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Friedman, J.H.: Multivariate Adaptive Regression Splines. Stanford linear accelerator center, Stanford, California, vol. 19, pp. 1–67 (1990)Google Scholar
  11. 11.
    Guttman, I.: Introductory Engineering Statistics. Wiley, Hoboken (1965)Google Scholar
  12. 12.
    Hartley, P.J., Judd, C.J.: Parametrization of Bézier-type B-spline curves and surfaces. Comput. Aided Des. 10, 130–134 (1978)CrossRefGoogle Scholar
  13. 13.
    Härdle, W.: Applied nonparametric regression 34(2), 341–342 (1989)Google Scholar
  14. 14.
    James, G.M.: Curve alignment by moments. Ann. Appl. Stat. 1, 480–501 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jekabsons, G.: ARESLab: Adaptive Regression Splines toolbox for Matlab/Octave. Riga Technical University, Riga, Latvia, Institute of Applied Computer Systems (2009)Google Scholar
  16. 16.
    Lakså, A.: Blending Technics for Curve and Surface Constructions. Narvik University College, Narvik (2012)Google Scholar
  17. 17.
    Shumaker, L.L.: Spline Functions, 3rd edn. Cambrige University Press, Cambridge (2007)CrossRefGoogle Scholar
  18. 18.
    Van der Walt, M.D.: Real-time, local spline interpolation schemes on bounded intervals. Appl. Math. Sci. 10, 205–234 (2015)Google Scholar
  19. 19.
    Vorontsov, K.V.: Lectures about algorithms for dependencies reconstruction (2007)Google Scholar
  20. 20.
    Weather service Norwegian Meteorological Institute and Norwegian Broadcasting Corporation, 2007–2017, 01.12.2016–03.01.2017.,
  21. 21.
    Zanaty, P., Dechevsky, L.T.: On the numerical performance of FEM based on piecewise rational smooth resolutions of unity on triangulations. In: AIP Conference Proceedings, vol. 1570, p. 191 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.R&D Group Simulations, Department of Computer Science and Computational Engineering, Faculty of Science and TechnologyUiT - The Arctic University of NorwayNarvikNorway

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