Discontinuity, Nonlinearity, and Complexity
High Degree Multivariate Fuzzy Approximation by QuasiInterpolation Neural Network Operators
Discontinuity, Nonlinearity, and Complexity 2(2) (2013) 125146  DOI:10.5890/DNC.2013.04.003
George A. Anastassiou
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A
Download Full Text PDF
Abstract
Here are considered in terms of multivariate fuzzy high approximation to the multivariate unit sequences of multivariate fuzzy quasiinterpolation neural network operators. These operators are multivariate fuzzy analogs of earlier considered multivariate real ones. The derived results generalize earlier real ones into the fuzzy setting. Here the high degree multivariate fuzzy pointwise and uniform convergence with rates to the multivariate fuzzy unit operator are given through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the N th order (N ≥ 1) Hfuzzy partial derivatives, of the involved multivariate fuzzy number valued function.
References

[1]  Wu, C. and Gong, Z. (2001), On Henstock integral of fuzzynumbervalued functions (I), Fuzzy Sets and Systems, 120(3), 523532. 

[2]  Goetschel Jr., R. and Voxman,W. (1986), Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, 3143. 

[3]  Wu, C. and Ma, M. (1991), On embedding problem of fuzzy number space: Part 1, Fuzzy Sets and Systems, 44, 3338. 

[4]  Anastassiou, G.A. (2004), Fuzzy approximation by fuzzy convolution type operators, Computers and Mathematics, 48, 13691386. 

[5]  Anastassiou, G.A. (2007), Fuzzy Korovkin Theorems and Inequalities, Journal of Fuzzy Mathematics, 15 (1), 169205. 

[6]  Kaleva, O.(1987), Fuzzy differential equations, Fuzzy Sets and Systems, 24, 301317. 

[7]  Chen, Z. and Cao, F. (2009), The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58, 758765. 

[8]  Anastassiou, G.A. (2011), Inteligent Systems: Approximation by Artificial Neural Networks, Springer, Heidelberg. 

[9]  Anastassiou, G.A. (2012), Univariate sigmoidal neural network approximation, Journal of Computational Analysis and Applications, 14(4), 659690. 

[10]  Anastassiou, G.A. (2011),Multivariate sigmoidal neural network approximation, Neural Networks, 2437824386. 

[11]  Anastassiou, G.A. (2011),Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics, 61, 809821. 

[12]  Anastassiou, G.A. (2011), Univariate hyperbolic tangent neural network approximation, Mathematics and Computer Modelling, 53, 11111132. 

[13]  Anastassiou, G.A. ( 2013), Approximation by Neural Network Iterates, in Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012, pp. 120, Editors: G. Anastassiou and O. Duman, Springer, New York. 

[14]  Mitchell, T.M. (1997), Machine Learning,WCBMcGrawHill, New York. 

[15]  McCulloch, W. and Pitts, W. (1943), A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 7, 115133. 

[16]  Kim, Y.K. and Ghil, B.M. (1997), Integrals of fuzzynumbervalued functions, Fuzzy Sets and Systems, 86, 213 222. 

[17]  Anastassiou, G.A.(2006),Higher order Fuzzy Korovkin Theory via inequalities, Communications in Applied Analysis, 10 ( 2), 359392. 