A bicubic model for local smoothing of surfaces is constructed on the base of pivot points. Such an approach allows reducing the dimension of matrix of normal equations more than twice. The model enables to increase essentially the speed and stability of calculations. The algorithms, constructed by the aid of the offered model, can be used both in applications and the development of global methods for smoothing and approximation of surfaces.
The paper deals with the recently proposed autotracking piecewise cubic approximation (APCA) based on the discrete projective transformation, and neural networks (NN). The suggested new approach facilitates the analysis of data with complex dependence and relatively small errors. We introduce a new representation of polynomials that can provide different local approximation models. We demonstrate how APCA can be applied to especially noisy data thanks to NN and local estimations. On the other hand, the new approximation method also has its impact on neural networks. We show how APCA helps to decrease the computation time of feed forward NN.
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≥2 reference points and decreases the polynomial degree by r−1. Then a general representation of polynomials is proposed based on r≥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 Tr(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.