Automatic Curve Fitting Based on Radial Basis Functions and a Hierarchical Genetic Algorithm

G Trejo-Caballero,C H Garcia-Capulin,O G Ibarra-Manzano,J G Avina-Cervantes,C Torres-Huitzil,H Rostro-Gonzalez

doi:10.1155/2015/731207

Abstract

Curve fitting is a very challenging problem that arises in a wide variety of scientific and engineering applications. Given a set of data points, possibly noisy, the goal is to build a compact representation of the curve that corresponds to the best estimate of the unknown underlying relationship between two variables. Despite the large number of methods available to tackle this problem, it remains challenging and elusive. In this paper, a new method to tackle such problem using strictly a linear combination of radial basis functions (RBFs) is proposed. To be more specific, we divide the parameter search space into linear and nonlinear parameter subspaces. We use a hierarchical genetic algorithm (HGA) to minimize a model selection criterion, which allows us to automatically and simultaneously determine the nonlinear parameters and then, by the least-squares method through Singular Value Decomposition method, to compute the linear parameters. The method is fully automatic and does not require subjective parameters, for example, smooth factor or centre locations, to perform the solution. In order to validate the efficacy of our approach, we perform an experimental study with several tests on benchmarks smooth functions. A comparative analysis with two successful methods based on RBF networks has been included.

Highlights

In the literature, there are many methods to tackle the curve fitting problem, which remains challenging and elusive
We present a comparison with the well-known methods based on radial basis neural networks (RBNN) and general regression neural networks (GRNN)
We have proposed an efficient hierarchical genetic algorithm to tackle the automatic curve fitting problem

Summary

Introduction

There are many methods to tackle the curve fitting problem, which remains challenging and elusive. The curve fitting problem has been mainly addressed by using typical methods based on linear models [7,8,9,10]. These methods consider that, given a set of data points, any function can be properly approximated on a specific interval using a linear combination of a set of m fixed functions often called basis functions. Where | ⋅ | denotes the norm used to measure the distance between any point x and the centre c of the basis function and φ(⋅) is a specific type of RBF.

Methods

Results

Conclusion