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.

[1]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[2]  Jun Wang,et al.  Fluctuation prediction of stock market index by Legendre neural network with random time strength function , 2012, Neurocomputing.

[3]  Rigos Anastasios,et al.  On the systematic implementation of artificial neural networks in the classification of variance images and shoreline extraction , 2014 .

[4]  J. Adamowski,et al.  A wavelet neural network conjunction model for groundwater level forecasting , 2011 .

[5]  David E. Crowley,et al.  Artificial neural network modeling of microbial community structures in the Atlantic Forest of Brazil , 2014 .

[6]  Limin Chen,et al.  A hybrid empirical-Bayesian artificial neural network model of salinity in the San Francisco Bay-Delta estuary , 2017, Environ. Model. Softw..

[7]  P. Cortez,et al.  A data mining approach to predict forest fires using meteorological data , 2007 .

[8]  Elizabeth E. Holmes,et al.  BEYOND THEORY TO APPLICATION AND EVALUATION: DIFFUSION APPROXIMATIONS FOR POPULATION VIABILITY ANALYSIS , 2004 .

[9]  Snehashish Chakraverty,et al.  Applied Soft Computing , 2016 .

[10]  Erik D. Goodman,et al.  A neighbor-based learning particle swarm optimizer with short-term and long-term memory for dynamic optimization problems , 2018, Inf. Sci..

[11]  Tsu-Tian Lee,et al.  The Chebyshev-polynomials-based unified model neural networks for function approximation , 1998, IEEE Trans. Syst. Man Cybern. Part B.

[12]  Dražen Lončar,et al.  Artificial neural network modelling approach for a biomass gasification process in fixed bed gasifiers. , 2014 .

[13]  J. Knouft,et al.  Climate and local abundance in freshwater fishes , 2016, Royal Society Open Science.

[14]  Chiara Polce,et al.  Predicting ground temperatures across European landscapes , 2015 .

[15]  Greet Janssens-Maenhout,et al.  Near-term projection of anthropogenic emission trends using neural networks , 2014 .

[16]  Marc Deconchat,et al.  Simple Neural Network Reveals Unexpected Patterns of Bird Species Richness in Forest Fragments , 2005, Landscape Ecology.

[17]  Ivan Maric,et al.  Optimization of self-organizing polynomial neural networks , 2013, Expert Syst. Appl..

[18]  Anastasios Rigos,et al.  A Chebyshev polynomial radial basis function neural network for automated shoreline extraction from coastal imagery , 2016, Integr. Comput. Aided Eng..

[19]  Najdan Vukovic,et al.  A comprehensive experimental evaluation of orthogonal polynomial expanded random vector functional link neural networks for regression , 2017, Appl. Soft Comput..

[20]  R. B. Jackson,et al.  Global biodiversity scenarios for the year 2100. , 2000, Science.

[21]  Zhenliang Liao,et al.  An optimization of artificial neural network model for predicting chlorophyll dynamics , 2017 .

[22]  Peter Coad,et al.  Proactive management of estuarine algal blooms using an automated monitoring buoy coupled with an artificial neural network , 2014, Environ. Model. Softw..

[23]  Despina Deligiorgi,et al.  Applying linear and nonlinear models for the estimation of particulate matter variability. , 2019, Environmental pollution.

[24]  LeeAnn Racz,et al.  Detecting recalcitrant organic chemicals in water with microbial fuel cells and artificial neural networks. , 2014, The Science of the total environment.

[25]  Ryo Shoji,et al.  Prediction of genotoxicity of various environmental pollutants by artificial neural network simulation , 2006, Molecular Diversity.

[26]  Bidyadhar Subudhi,et al.  A differential evolution based neural network approach to nonlinear system identification , 2011, Appl. Soft Comput..

[27]  Nikolaos Mitianoudis,et al.  A Hermite neural network incorporating artificial bee colony optimization to model shoreline realignment at a reef-fronted beach , 2017, Neurocomputing.

[28]  Tyler Wagner,et al.  A regional neural network ensemble for predicting mean daily river water temperature , 2014 .

[29]  Athanasios Tsanas,et al.  Accurate quantitative estimation of energy performance of residential buildings using statistical machine learning tools , 2012 .

[30]  George E. Tsekouras,et al.  Modeling Beach Rotation Using a Novel Legendre Polynomial Feedforward Neural Network Trained by Nonlinear Constrained Optimization , 2016, AIAI.

[31]  Gary R. Weckman,et al.  Modeling microalgal abundance with artificial neural networks: Demonstration of a heuristic 'Grey-Box' to deconvolve and quantify environmental influences , 2012, Environ. Model. Softw..

[32]  Yu Liu,et al.  A hybrid differential evolution algorithm for mixed-variable optimization problems , 2018, Inf. Sci..

[33]  Sung-Kwun Oh,et al.  Design of K-means clustering-based polynomial radial basis function neural networks (pRBF NNs) realized with the aid of particle swarm optimization and differential evolution , 2012, Neurocomputing.

[34]  Jun Zhu,et al.  Modeling and inference of animal movement using artificial neural networks , 2011, Environmental and Ecological Statistics.

[35]  Viktor Pocajt,et al.  A linear and non-linear polynomial neural network modeling of dissolved oxygen content in surface water: Inter- and extrapolation performance with inputs' significance analysis. , 2018, The Science of the total environment.

[36]  James K. Lein,et al.  Implementing remote sensing strategies to support environmental compliance assessment: A neural network application , 2009 .

[37]  Despina Deligiorgi,et al.  Spatial estimation of urban air pollution with the use of artificial neural network models , 2018, Atmospheric Environment.

[38]  María José del Jesús,et al.  KEEL: a software tool to assess evolutionary algorithms for data mining problems , 2008, Soft Comput..

[39]  R. Storn,et al.  Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series) , 2005 .

[40]  Sotiris Papantoniou,et al.  Prediction of outdoor air temperature using neural networks: Application in 4 European cities , 2016 .

[41]  Alex ChiChung Kot,et al.  Nonlinear dynamic system identification using Chebyshev functional link artificial neural networks , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[42]  Indra Narayan Kar,et al.  On-line system identification of complex systems using Chebyshev neural networks , 2007, Appl. Soft Comput..

[43]  Khashayar Khorasani,et al.  Constructive feedforward neural networks using Hermite polynomial activation functions , 2005, IEEE Transactions on Neural Networks.

[44]  Ke Chen,et al.  A hybrid particle swarm optimizer with sine cosine acceleration coefficients , 2018, Inf. Sci..

[45]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[46]  A. G. Ivakhnenko,et al.  Polynomial Theory of Complex Systems , 1971, IEEE Trans. Syst. Man Cybern..

[47]  Andréa Oliveira Souza da Costa,et al.  Use of neural networks for monitoring surface water quality changes in a neotropical urban stream. , 2009, Environmental monitoring and assessment.

[48]  Aaron C. Zecchin,et al.  Selection of smoothing parameter estimators for general regression neural networks - Applications to hydrological and water resources modelling , 2014, Environ. Model. Softw..

[49]  Gary R. Weckman,et al.  Modeling net ecosystem metabolism with an artificial neural network and Bayesian belief network , 2011, Environ. Model. Softw..

[50]  M. Kocsis,et al.  IMPACTS OF CLIMATE CHANGE ON LEPIDOPTERA SPECIES AND COMMUNITIES , 2011 .

[51]  S. Gyamfi,et al.  Residential peak electricity demand response—Highlights of some behavioural issues , 2013 .

[52]  D. Borchardt,et al.  Bioindication of chemical and hydromorphological habitat characteristics with benthic macro-invertebrates based on Artificial Neural Networks , 2001, Aquatic Ecology.