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Keywords:
polynomial representation; approximation model; smooth connection; consistency; asymptotic normality
polynomial representation; approximation model; smooth connection; consistency; asymptotic normality
Summary:
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 $r \geq 2$ reference points and decreases the polynomial degree by $r-1$. Then a general representation of polynomials is proposed based on $r \geq 1$ 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.
References:
[1] Csörgő, S., Mielniczuk, J.: Nonparametric regression under long-range dependent normal errors. Ann. Statist. 23 (1995), 3, 1000-1014. DOI 10.1214/aos/1176324633 | MR 1345211 | Zbl 0852.62035
[2] Dikoussar, N. D.: Function parametrization by using 4-point transforms. Comput. Phys. Commun. 99 (1997), 235-254. DOI 10.1016/S0010-4655(96)00110-5 | Zbl 0927.65009
[3] Dikoussar, N. D.: Kusochno-kubicheskoje priblizhenije I sglazhivanije krivich v rezhime adaptacii. Communication JINR, P10-99-168, Dubna 1999.
[4] Dikusar, N. D.: The basic element method. Math. Models Comput. Simulat. 3 (2011), 4, 492-507. DOI 10.1134/S2070048211040053 | MR 2810219 | Zbl 1240.65077
[5] Dikoussar, N. D., Török, Cs.: Automatic knot finding for piecewise-cubic approximation. Matem. Mod. 18 (2006), 3, 23-40. MR 2255951 | Zbl 1099.65014
[6] Dikoussar, N. D., Török, Cs.: On one approach to local surface smoothing. Kybernetika 43 (2007), 4, 533-546. MR 2377931 | Zbl 1139.65009
[7] Eubank, R. L.: Nonparametric Regression and Spline Smoothing. Marcel Dekker, Inc., 1999. MR 1680784 | Zbl 0936.62044
[8] Kahaner, D., Moler, C., Nash, S.: Numerical Methods and Software. Prentice-Hall, Inc., 1989. Zbl 0744.65002
[9] Kepič, T., Török, Cs., Dikoussar, N. D.: Wavelet compression. In: 13. International Workshop on Computational Statistics, Bratislava 2004, pp. 49-52.
[10] Matejčiková, A., Török, Cs.: Noise suppression in RDPT. Forum Statisticum Slovacum 3 (2005), 199-203.
[11] Nadaraya, E. A.: On estimating regression. Theory Probab. Appl. 9 (1964), 141-142. Zbl 0136.40902
[12] Révayová, M., Török, Cs.: Piecewise approximation and neural networks. Kybernetika 43 (2007), 4, 547-559. MR 2377932 | Zbl 1145.68495
[13] Révayová, M., Török, Cs.: Reference points based recursive approximation. Kybernetika 49 (2013), 1, 60-72.
[14] Riplay, B. D.: Pattern Recognition and Neural Networks. Cambridge University Press 1996. MR 1438788
[15] Seber, G. A. F.: Linear Regression Analysis. J. Wiley and Sons, New York 1977. MR 0436482 | Zbl 1029.62059
[16] Török, Cs.: 4-point transforms and approximation. Comput. Phys. Commun. 125 (2000), 154-166. DOI 10.1016/S0010-4655(99)00483-X | Zbl 0976.65011
[17] Török, Cs., Dikoussar, N. D.: Approximation with DPT. Comput. Math. Appl. 38 (1999), 211-220. DOI 10.1016/S0898-1221(99)00276-X | MR 1718884
[18] Trefethen, L. N.: Approximation Theory and Approximation Practice. SIAM, 2013. MR 3012510 | Zbl 1264.41001
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