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The Optimal Shape Parameter for the Least Squares Approximation Based on the Radial Basis Function

Author

Listed:
  • Sanpeng Zheng

    (School of Mathematics Science & Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beihang University, Beijing 100191, China)

  • Renzhong Feng

    (School of Mathematics Science & Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beihang University, Beijing 100191, China)

  • Aitong Huang

    (School of Mathematics Science & Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beihang University, Beijing 100191, China)

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.

Suggested Citation

  • Sanpeng Zheng & Renzhong Feng & Aitong Huang, 2020. "The Optimal Shape Parameter for the Least Squares Approximation Based on the Radial Basis Function," Mathematics, MDPI, vol. 8(11), pages 1-20, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1923-:d:438753
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    References listed on IDEAS

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    1. Christopher M. Cotnoir & Balša Terzić, 2017. "Decoupling linear and nonlinear regimes: an evaluation of efficiency for nonlinear multidimensional optimization," Journal of Global Optimization, Springer, vol. 68(3), pages 663-675, July.
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    Cited by:

    1. Zheng, Sanpeng & Feng, Renzhong, 2023. "A variable projection method for the general radial basis function neural network," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    2. Li, Yang & Liu, Dejun & Yin, Zhexu & Chen, Yun & Meng, Jin, 2023. "Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 440(C).

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