On training non-uniform fuzzy partitions for function approximation using differential evolution: A study on fuzzy transform and fuzzy projection

Abstract

This paper focuses on the use of differential evolution to improve the approximation properties of function approximation models based on fuzzy partitions. Two cases are considered: Fuzzy transform and Fuzzy projection, and the design of hybrid evolutionary fuzzy systems, is studied. Even though function approximation techniques based on fuzzy partitions have been well studied, few papers consider the problem of centroid selection of the basis functions. Thus, in most cases uniform fuzzy partitions are considered. By using an evolutionary algorithm a systematic approach on the selection of the partition, is provided. The optimisation problem involves the determination of the model parameters, which in our case are the fuzzy partition’s membership functions’ locations. The proposed method is tested on the scattered data approximation problem, in a regression sense, that is given a set of sparse data the latent function is approximated. Numerical studies on one and two-dimensional test functions demonstrate that the evolutionary algorithm based fuzzy projection displays high performance in terms of approximation error. Moreover, the proposed approach shows high approximation capabilities with a small number of basis functions. Comparison results, with uniform fuzzy partition models, neural networks and support vector machines, are provided.