Hybrid adaptive wavelet-neuro-fuzzy system for chaotic time series identification Y. Bodyanskiy ⇑ , O. Vynokurova Control Systems Research Laboratory, Kharkiv National University of Radio Electronics, Lenina av. 14, Kharkiv 61166, Ukraine article info Article history: Available online 2 August 2012 Keywords: Hybrid adaptive wavelet-neuro-fuzzy system Learning algorithm Chaotic time series Identification Prediction abstract In the paper a five-layers architecture of hybrid wavelet-neuro-fuzzy system which is using the adaptive W-neurons as the nodes is proposed. W-neuron is a neuron which structure is similar to a radial basis functions network, but instead of conventional radial basis func- tions we used multidimensional adaptive wavelet activation–membership functions. The distinctive feature of the proposed system is usage of the wavelets as membership func- tions in the antecedent layer, and the adaptive multidimensional wavelets as activation functions in the consequent layer that can tune not only the dilation and translation parameters but also its own form during the learning process. The learning algorithms for all antecedent and consequent functions’ parameters that have both following and fil- tering properties are proposed. The experimental results have shown that this wavelet- neuro-fuzzy system has improved approximation properties and has a higher learning rate in comparison with usual wavelet-neuro-fuzzy networks. The proposed hybrid wavelet- neuro-fuzzy system can be used to solve tasks of diagnosis, forecasting, emulation, and identification of nonlinear chaotic and stochastic non-stationary processes. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Nowadays the methods of computational intelligence and soft computing are widely used for solving chaotic signals pre- diction, identification, and emulation [1–4]. Such methods allow solving a large class of information processing problems and uppermost, identification, intelligent control tasks, arbitrary time series prediction under structural and parametrical uncertainty conditions. The multilayer feedforward networks are most known and popular. The efficiency of multilayer networks is explained by their universal approximation properties in aggregate with relative compact presentation of the simulated non-linear sys- tem. It means that they can be successfully used in non-linear systems identification tasks. Though the principal disadvan- tage of the multilayer networks is a low learning rate based on backpropagation algorithm which makes it impossible to use them in real time tasks. For last years also the neuro-fuzzy systems have been an increasingly popular technique of soft computing [5–13] suc- cessfully applied for the processing of information containing complex nonlinear regularities and distortions of all kinds. Along with neuro-fuzzy systems for processing signals of all kinds, the wavelet transform is widely used [14–18]. The com- putational advantages of wavelet functions are the following [14–18]: wavelets have local support and provide compact lo- cal representation of signals both in frequency and in time domain; wavelets have scaling (width) and shifting parameters (centers), that allows to process signals with local features. 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2012.07.044 ⇑ Corresponding author. E-mail addresses: bodya@kture.kharkov.ua (Y. Bodyanskiy), vinokurova@kture.kharkov.ua (O. Vynokurova). Information Sciences 220 (2013) 170–179 Contents lists available at SciVerse ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins