Abstract
The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. The RBF is especially suitable for scattered data approximation and high dimensional function approximation. The smoothness and approximation accuracy of the RBF are affected by its shape parameter. There has been some research on the shape parameter, but the research on the optimal shape parameter of the least squares based on the RBF is scarce. This paper proposes a way for the measurement of the optimal shape parameter of the least squares approximation based on the RBF and an algorithm to solve the corresponding optimal parameter. The method consists of considering the shape parameter as an optimization variable of the least squares problem, such that the linear least squares problem becomes nonlinear. A dimensionality reduction is applied to the nonlinear least squares problem in order to simplify the objective function. To solve the optimization problem efficiently after the dimensional reduction, the derivative-free optimization is adopted. The numerical experiments indicate that the proposed method is efficient and reliable. Multiple kinds of RBFs are tested for their effects and compared. It is found through the experiments that the RBF least squares with the optimal shape parameter is much better than the polynomial least squares. The method is successfully applied to the fitting of real data.
Highlights
The radial basis function (RBF) is a class of approximation functions that are widely used nowadays in many fields
We think that the curved surface fitting by the RBF least squares with the optimal shape parameter is better than that by the least squares based on polynomial bases
This paper investigates the selection of shape parameters in the RBF least squares
Summary
The radial basis function (RBF) is a class of approximation functions that are widely used nowadays in many fields. Majdisova and Skala [9,10] examined the RBF least squares with real data and synthetic data They tried to improve the approximation by changing the shape parameters and defined the optimal shape parameters by abundant experiments. Rippa [11] employed the idea of cross validation in statistics and constructed the cost function for the RBF interpolation He obtained the optimal shape parameter by minimizing the cost function. This method is known as the leave-one-out cross validation method All of the above studies proposed a way of measuring the optimal shape parameters for their respective research problems and provided a method to obtain those parameters. This paper focuses on how to define a method of measuring the optimal shape parameter in RBF least squares approximation and to obtain that parameter efficiently.
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