Universal Approximation of a Class of Interval Type-2 Fuzzy Neural Networks in Nonlinear Identification

Neural networks (NNs), type-1 fuzzy logic systems (T1FLSs), and interval type-2 fuzzy logic systems (IT2FLSs) have been shown to be universal approximators, which means that they can approximate any nonlinear continuous function. Recent research shows that embedding an IT2FLS on an NN can be very effective for a wide number of nonlinear complex systems, especially when handling imperfect or incomplete information. In this paper we show, based on the Stone-Weierstrass theorem, that an interval type-2 fuzzy neural network (IT2FNN) is a universal approximator, which uses a set of rules and interval type-2membership functions (IT2MFs) for this purpose. Simulation results of nonlinear function identification using the IT2FNN for one and three variables and for the Mackey-Glass chaotic time series prediction are presented to illustrate the concept of universal approximation.

[1]  Shigeo Abe,et al.  Neural Networks and Fuzzy Systems , 1996, Springer US.

[2]  Jin S. Lee,et al.  Universal approximation by hierarchical fuzzy system with constraints on the fuzzy rule , 2002, Fuzzy Sets Syst..

[3]  Juan R. Castro,et al.  Hybrid Learning Algorithm for Interval Type-2 Fuzzy Neural Networks , 2007 .

[4]  Vladik Kreinovich,et al.  Fuzzy Rule Based Modeling as a Universal Approximation Tool , 1998 .

[5]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[6]  Yücel Koçyigit,et al.  Differentiating types of muscle movements using a wavelet based fuzzy clustering neural network , 2009, Expert Syst. J. Knowl. Eng..

[7]  Narges Shafaei Bajestani,et al.  Application of optimized Type 2 fuzzy time series to forecast Taiwan stock index , 2009, 2009 2nd International Conference on Computer, Control and Communication.

[8]  Jerry M. Mendel,et al.  Generating fuzzy rules by learning from examples , 1992, IEEE Trans. Syst. Man Cybern..

[9]  Faa-Jeng Lin,et al.  Adaptive Control of Two-Axis Motion Control System Using Interval Type-2 Fuzzy Neural Network , 2009, IEEE Transactions on Industrial Electronics.

[10]  L X Wang,et al.  Fuzzy basis functions, universal approximation, and orthogonal least-squares learning , 1992, IEEE Trans. Neural Networks.

[11]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[12]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[13]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[14]  W. Pedrycz,et al.  OR/AND neuron in modeling fuzzy set connectives , 1994, IEEE Trans. Fuzzy Syst..

[15]  R. E. Edwards,et al.  Functional Analysis: Theory and Applications , 1965 .

[16]  Sergios Theodoridis,et al.  Pattern Recognition, Third Edition , 2006 .

[17]  Oscar Castillo,et al.  A Hybrid Learning Algorithm for Interval Type-2 Fuzzy Neural Networks: The Case of Time Series Prediction , 2008, Soft Computing for Hybrid Intelligent Systems.

[18]  James J. Buckley,et al.  Universal fuzzy controllers , 1992, Autom..

[19]  Bart Kosko,et al.  Fuzzy Systems as Universal Approximators , 1994, IEEE Trans. Computers.

[20]  Lida Xu,et al.  A new type of recurrent fuzzy neural network for modeling dynamic systems , 2001, Knowl. Based Syst..

[21]  Jyh-Shing Roger Jang,et al.  ANFIS: adaptive-network-based fuzzy inference system , 1993, IEEE Trans. Syst. Man Cybern..

[22]  Lida Xu,et al.  Dynamic recurrent neural networks for a hybrid intelligent decision support system for the metallurgical industry , 1999, Expert Syst. J. Knowl. Eng..

[23]  Lida Xu,et al.  A variational approach to intensity approximation for remote sensing images using dynamic neural networks , 2003, Expert Syst. J. Knowl. Eng..

[24]  Jerry M. Mendel,et al.  Interval type-2 fuzzy logic systems , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[25]  Rahime Ceylan,et al.  Classification of ECG arrhythmias using Type-2 Fuzzy Clustering Neural Network , 2009, 2009 14th National Biomedical Engineering Meeting.

[26]  Saeed Panahian Fard,et al.  Interval type-2 fuzzy neural networks version of the Stone-Weierstrass theorem , 2011, Neurocomputing.

[27]  John T. Rickard,et al.  Fuzzy Subsethood for Fuzzy Sets of Type-2 and Generalized Type- ${n}$ , 2009, IEEE Transactions on Fuzzy Systems.

[28]  E. Mizutani,et al.  Neuro-Fuzzy and Soft Computing-A Computational Approach to Learning and Machine Intelligence [Book Review] , 1997, IEEE Transactions on Automatic Control.

[29]  Ching-Hung Lee,et al.  A recurrent interval type-2 fuzzy neural network with asymmetric membership functions for nonlinear system identification , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[30]  W. Rudin.  Principles of mathematical analysis , 1964 .

[31]  James J. Buckley,et al.  Can fuzzy neural nets approximate continuous fuzzy functions , 1994 .

[32]  A. H. Stone,et al.  Products of nearly compact spaces , 1966 .

[33]  J. Buckley.  Sugeno type controllers are universal controllers , 1993 .

[34]  Oscar Castillo,et al.  A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks , 2009, Inf. Sci..

[35]  Ching-Hung Lee,et al.  TYPE-2 FUZZY NEURAL NETWORK SYSTEMS AND LEARNING , 2002 .

[36]  Robert Ivor John,et al.  On Constructing Parsimonious Type-2 Fuzzy Logic Systems via Influential Rule Selection , 2009, IEEE Transactions on Fuzzy Systems.

[37]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[38]  Chi-Hsu Wang,et al.  Dynamical optimal training for interval type-2 fuzzy neural network (T2FNN) , 2003, SMC'03 Conference Proceedings. 2003 IEEE International Conference on Systems, Man and Cybernetics. Conference Theme - System Security and Assurance (Cat. No.03CH37483).