Kybernetika 49 no. 4, 644-662, 2013

Reference points based transformation and approximation

Csaba Török


Interpolating and approximating polynomials have been living separately more than two centuries. Our aim is to propose a general parametric regression model that incorporates both interpolation and approximation. The paper introduces first a new $r$-point transformation that yields a function with a simpler geometrical structure than the original function. It uses $r\ge 2$ reference points and decreases the polynomial degree by $r-1$. Then a general representation of polynomials is proposed based on $r\ge 1$ reference points. The two-part model, which is suited to piecewise approximation, consist of an ordinary least squares polynomial regression and a reparameterized one. The later is the central component where the key role is played by the reference points. It is constructed based on the proposed representation of polynomials that is derived using the $r$-point transformation $T_r(x)$. The resulting polynomial passes through $r$ reference points and the other points approximates. Appropriately chosen reference points ensure quasi smooth transition between the two components and decrease the dimension of the LS normal matrix. We show that the model provides estimates with such statistical properties as consistency and asymptotic normality.


asymptotic normality, consistency, approximation model, polynomial representation, smooth connection


41A10, 65D05, 65D07, 65D10, 62J05, 62F12


  1. S. Csörgő and J. Mielniczuk: Nonparametric regression under long-range dependent normal errors. Ann. Statist. 23 (1995), 3, 1000-1014.   CrossRef
  2. N. D. Dikoussar: Function parametrization by using 4-point transforms. Comput. Phys. Commun. 99 (1997), 235-254.   CrossRef
  3. N. D. Dikoussar: Kusochno-kubicheskoje priblizhenije I sglazhivanije krivich v rezhime adaptacii. Communication JINR, P10-99-168, Dubna 1999.   CrossRef
  4. N. D. Dikusar: The basic element method. Math. Models Comput. Simulat. 3 (2011), 4, 492-507.   CrossRef
  5. N. D. Dikoussar and Cs. Török: Automatic knot finding for piecewise-cubic approximation. Matem. Mod. 18 (2006), 3, 23-40.   CrossRef
  6. N. D. Dikoussar and Cs. Török: On one approach to local surface smoothing. Kybernetika 43 (2007), 4, 533-546.   CrossRef
  7. R. L. Eubank: Nonparametric Regression and Spline Smoothing. Marcel Dekker, Inc., 1999.   CrossRef
  8. D. Kahaner, C. Moler and S. Nash: Numerical Methods and Software. Prentice-Hall, Inc., 1989.   CrossRef
  9. T. Kepič, Cs. Török and N. D. Dikoussar: Wavelet compression. In: 13. International Workshop on Computational Statistics, Bratislava 2004, pp. 49-52.   CrossRef
  10. A. Matejčiková and Cs. Török: Noise suppression in RDPT. Forum Statisticum Slovacum 3 (2005), 199-203.   CrossRef
  11. E. A. Nadaraya: On estimating regression. Theory Probab. Appl. 9 (1964), 141-142.   CrossRef
  12. M. Révayová and Cs. Török: Piecewise approximation and neural networks. Kybernetika 43 (2007), 4, 547-559.   CrossRef
  13. M. Révayová and Cs. Török: Reference points based recursive approximation. Kybernetika 49 (2013), 1, 60-72.   CrossRef
  14. B. D. Riplay: Pattern Recognition and Neural Networks. Cambridge University Press 1996.   CrossRef
  15. G. A. F. Seber: Linear Regression Analysis. J. Wiley and Sons, New York 1977.   CrossRef
  16. Cs. Török: 4-point transforms and approximation. Comput. Phys. Commun. 125 (2000), 154-166.   CrossRef
  17. Cs. Török and N. D. Dikoussar: Approximation with DPT. Comput. Math. Appl. 38 (1999), 211-220.   CrossRef
  18. L. N. Trefethen: Approximation Theory and Approximation Practice. SIAM, 2013.   CrossRef