Machine learning regression based on particle bernstein polynomials for nonlinear system identification

Polynomials have shown to be useful basis functions in the identification of nonlinear systems. However estimation of the unknown coefficients requires expensive algorithms, as for instance it occurs by applying an optimal least square approach. Bernstein polynomials have the property that the coefficients are the values of the function to be approximated at points in a fixed grid, thus avoiding a time-consuming training stage. This paper presents a novel machine learning approach to regression, based on new functions named particle-Bernstein polynomials, which is particularly suitable to solve multivariate regression problems. Several experimental results show the validity of the technique for the identification of nonlinear systems and the better performance achieved with respect to the standard techniques.

1. Introduction

The goal on nonlinear system identification is to estimate a relation between inputs and outputs generated by an unknown dynamical system [1]. As this problem is equivalent to derive a model given a set of input-output data, in the language of machine learning it falls within the problem of supervised learning. Over the years a large variety of different approaches has been proposed in the literature to face this problem [2]. One of the most popular is the Lee-Schetzen method that identifies the Volterra kernels of nonlinear systems stimulated by random inputs with assigned statistic [3], [4]. Simplified Volterra-based models which combine a static nonlinearity and a linear dynamical system (Hammerstein-Wiener systems) have been profitably used to overcome calculation of multidimensional Volterra kernels [5], [7]. Because of nonlinear signal processing and learning capability, artificial neural networks (ANN's) have become a powerful tool for nonlinear system identification [8], [9]. Recently machine learning techniques such as support vector machine (SVM) are progressing rapidly, and overcomes the neural networks' shortcomings, that is local minimizing and inadequacy to statistical problems [10], [11].