A Chebyshev polynomial feedforward neural network trained by differential evolution and its application in environmental case studies

Abstract

This paper introduces a polynomial feedforward neural network based on Chebyshev polynomials able to effectively model non-linear and highly complex environmental data. The data sets were cautiously selected from the fields of biology, ecology, climate, and environmental management, and economics as to represent a scientifically meaningful and consistent corpus of disparate domains of intensive focus and interest in current ecological/environmental research, covering issues related to body growth/age, biomass production, energy efficiency/consumption, and ecology/geographic extension. The proposed network uses a number of layers to estimate the output in terms of a weighted sum of truncated Chebyshev series expansions applied to linear combinations of the input variables, and it is trained by the differential evolution algorithm. Its performance was compared to three neural networks. First, a polynomial feedforward network that uses Hermite polynomials as activation function in the hidden nodes; second, a radial basis function neural network; third, a Takagi-Sugeno-Kang neuro-fuzzy network. All the above networks were trained by evolutionary optimization algorithms. The comparison was carried out by standard criteria such as the root mean square error and the mean absolute error. Moreover, a non-parametric Kruskal-Wallis statistical test used to compare the median values of the root mean square errors between methods. The main experimental outcomes are: (a) the network's efficiency improves for higher polynomial orders, (b) the statistical analysis suggests that the proposed network appears to be very competitive to the other three networks.