Mathematical Problems in Engineering

Volume 2015, Article ID 731207, 14 pages

http://dx.doi.org/10.1155/2015/731207

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

^{1}Department of Electronics, DICIS-University of Guanajuato, Comunidad de Palo Blanco s/n, 36885 Salamanca, GTO, Mexico^{2}Department of Mechatronics, ITESI, Carretera Irapuato-Silao Km. 12.5, 36821 Irapuato, GTO, Mexico^{3}Information Technology Laboratory, CINVESTAV, Parque Científico y Tecnológico, 87130 Ciudad Victoria, TAMPS, Mexico

Received 30 September 2015; Revised 21 November 2015; Accepted 7 December 2015

Academic Editor: Akemi Gálvez

Copyright © 2015 G. Trejo-Caballero et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

In the literature, there are many methods to tackle the curve fitting problem, which remains challenging and elusive. In this study, the goal is to build a compact representation of the curve that corresponds to the best estimate of the unknown relationship between two variables from a set of data points. Curve fitting is a fundamental tool in scientific and engineering applications such as system identification, data analysis and visualization, geometric modeling, CAD/CAM systems, medical imaging, and reverse engineering [1–6].

The curve fitting problem has been mainly addressed by using typical methods based on linear models [7–10]. 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 fixed functions often called basis functions. The main basis functions used to address the curve fitting problem are polynomials, piecewise polynomials (splines), and radial basis functions (RBFs). RBFs have typically shown a successful performance in methods based on interpolation, such as in [11, 12].

In [13], the authors present a method for spike classification enhancement based on the 3-Gaussian model fitting. In [14], the peak wavelength detection accuracy in fiber Bragg grating sensors is improved by using a wavelet filter and curve fitting based on RBF. In [15], it is demonstrated that a Gaussian curve fitting substantially reduces the measurement errors of spectrally distorted FBG sensors by employing a binary search algorithm that calculates and compares MSE values at only logically selected positions. In [12] the authors propose a method based on RBF interpolation to subpixel mapping of remote sensing images by fully exploiting the spatial information in the input images. The method uses RBF interpolation to predict the soft values at each subpixel. However, the interpolation methods present some limitations, especially when the number of data samples increases or the set of data samples is disturbed by noise. Thus, in order to solve the limitations of the interpolation methods, other methods have been proposed based on regression techniques. Nevertheless, these methods require the specification of global parameters such as the number of RBFs or centre locations to perform their solution. If this is not the case, if the basis functions and any parameters which they might contain change, then the model is nonlinear and the adequate choice of parameters becomes a continuous multimodal and multivariate nonlinear optimization problem, which must be addressed using modern techniques of Computational Intelligence.

In [16] the authors propose a method based on curve fitting with Gaussian functions for modeling carotid and radial artery pulse pressure waveforms. The method uses a fixed number of Gaussian functions, and their parameters are determined by using an algorithm based on particle swarm. Neural networks based on RBF have been applied to curve fitting in [17]. This paper considers fitting noisy curves using two neural networks: the multilayer feed forward network and the radial basis function network. A comparative analysis shows that when the noise level increases, the RBF networks are best suited for the reconstruction of noisy curves. However, most of these approaches use parameter values predefined without any justification, that is, in an empirical way. In [18], the spread parameter value of a neural network is selected by trial and error from a reasonably small interval previously determined.

It is found that RBFs have rarely been applied to curve fitting by using a linear combination to change the number of radial basis functions and simultaneously to optimize their parameters. This paper elucidates the feasibility of using RBFs in a linear combination to fit a set of data points disturbed by noise; that is, we propose a new method to tackle the curve fitting problem using a strictly linear combination of RBFs. The proposed approach divides the parameter search space into linear and nonlinear parameter subspaces. We use a hierarchical genetic algorithm to minimize a model selection criterion, which allows us to automatically and simultaneously determine the nonlinear parameters: spread, centre locations, and number of RBFs. The coefficients of the linear model (linear parameters) are computed solving a set of linear equations using the least-squares method through Singular Value Decomposition (SVD) method. In order to validate the efficacy of the proposed approach, we perform an experimental study with several tests on smooth functions benchmarks. The comparative analysis with two successful methods based on RBF networks has been included.

#### 2. Background

##### 2.1. Radial Basis Functions

RBFs are a particular class of functions whose value depends only on the distance from the centre. Specifically, their response decreases (*local response*) or increases (*global response*) monotonically with respect to the distance from a central point. In general, a radial basis function is represented by the following equation:where denotes the norm used to measure the distance between any point and the centre of the basis function and is a specific type of RBF. Usually, the norm used is Euclidean distance [19], and the types of RBFs commonly used are expressed in Table 1, where .