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Learning in a Time-Varying Environment by Making Use of the Stochastic Approximation and Orthogonal Series-Type Kernel Probabilistic Neural Network

  • Jacek M. Zurada
  • Maciej Jaworski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7203)

Abstract

In the paper stochastic approximation, in combining with general regression neural network, is applied for learning in a time-varying environment. The orthogonal-type kernel is applied to design the general regression neural networks. Sufficient conditions for weak convergence are given and simulation results are presented.

Keywords

IEEE Transaction Probabilistic Neural Network General Regression Neural Network Orthogonal Series Multiple Fourier Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    Alexits, G.: Convergence Problems of Orthogonal Series, Budapest, Academia and Kiado, pp. 261–264 (1961)Google Scholar
  2. 2.
    Bilski, J., Rutkowski, L.: A fast training algorithm for neural networks. IEEE Transactions on Circuits and Systems II 45, 749–753 (1998)CrossRefGoogle Scholar
  3. 3.
    Cierniak, R., Rutkowski, L.: On image compression by competitive neural networks and optimal linear predictors. Signal Processing: Image Communication - a Eurasip Journal 15(6), 559–565 (2000)CrossRefGoogle Scholar
  4. 4.
    Dvoretzky, A.: On stochastic approximation. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 39–56. University of California Press (1956)Google Scholar
  5. 5.
    Gałkowski, T., Rutkowski, L.: Nonparametric recovery of multivariate functions with applications to system identification. Proceedings of the IEEE 73, 942–943 (1985)CrossRefGoogle Scholar
  6. 6.
    Gałkowski, T., Rutkowski, L.: Nonparametric fitting of multivariable functions. IEEE Transactions on Automatic Control AC-31, 785–787 (1986)CrossRefGoogle Scholar
  7. 7.
    Greblicki, W., Rutkowska, D., Rutkowski, L.: An orthogonal series estimate of time-varying regression. Annals of the Institute of Statistical Mathematics 35, Part A, 147–160 (1983)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Greblicki, W., Rutkowski, L.: Density-free Bayes risk consistency of nonparametric pattern recognition procedures. Proceedings of the IEEE 69(4), 482–483 (1981)CrossRefGoogle Scholar
  9. 9.
    Nowicki, R.: Rough Sets in the Neuro-Fuzzy Architectures Based on Monotonic Fuzzy Implications. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 510–517. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Nowicki, R.: Nonlinear modelling and classification based on the MICOG defuzzifications. Journal of Nonlinear Analysis, Series A: Theory, Methods and Applications 7(12), 1033–1047 (2009)CrossRefGoogle Scholar
  11. 11.
    Patan, K., Patan, M.: Optimal Training Strategies for Locally Recurrent Neural Networks. Journal of Artificial Intelligence and Soft Computing Research 1(2), 103–114 (2011)Google Scholar
  12. 12.
    Robbins, H., Monro, S.: A stochastics approximation method. Annals Mathematics of Statistics 22, 400–407 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rutkowski, L.: Sequential estimates of probability densities by orthogonal series and their application in pattern classification. IEEE Transactions on Systems, Man, and Cybernetics SMC-10(12), 918–920 (1980)MathSciNetGoogle Scholar
  14. 14.
    Rutkowski, L.: Sequential estimates of a regression function by orthogonal series with applications in discrimination, New York, Heidelberg, Berlin. Lectures Notes in Statistics, vol. 8, pp. 236–244 (1981)Google Scholar
  15. 15.
    Rutkowski, L.: On system identification by nonparametric function fitting. IEEE Transactions on Automatic Control AC-27, 225–227 (1982)CrossRefGoogle Scholar
  16. 16.
    Rutkowski, L.: Orthogonal series estimates of a regression function with applications in system identification. In: Probability and Statistical Inference, pp. 343–347. D. Reidel Publishing Company, Dordrecht (1982)CrossRefGoogle Scholar
  17. 17.
    Rutkowski, L.: On Bayes risk consistent pattern recognition procedures in a quasi-stationary environment. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-4(1), 84–87 (1982)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rutkowski, L.: On-line identification of time-varying systems by nonparametric techniques. IEEE Transactions on Automatic Control AC-27, 228–230 (1982)CrossRefGoogle Scholar
  19. 19.
    Rutkowski, L.: On nonparametric identification with prediction of time-varying systems. IEEE Transactions on Automatic Control AC-29, 58–60 (1984)CrossRefGoogle Scholar
  20. 20.
    Rutkowski, L.: Nonparametric identification of quasi-stationary systems. Systems and Control Letters 6, 33–35 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Rutkowski, L.: The real-time identification of time-varying systems by nonparametric algorithms based on the Parzen kernels. International Journal of Systems Science 16, 1123–1130 (1985)zbMATHCrossRefGoogle Scholar
  22. 22.
    Rutkowski, L.: A general approach for nonparametric fitting of functions and their derivatives with applications to linear circuits identification. IEEE Transactions Circuits Systems CAS-33, 812–818 (1986)CrossRefGoogle Scholar
  23. 23.
    Rutkowski, L.: Sequential pattern recognition procedures derived from multiple Fourier series. Pattern Recognition Letters 8, 213–216 (1988)zbMATHCrossRefGoogle Scholar
  24. 24.
    Rutkowski, L.: Nonparametric procedures for identification and control of linear dynamic systems. In: Proceedings of 1988 American Control Conference, June 15-17, pp. 1325–1326 (1988)Google Scholar
  25. 25.
    Rutkowski, L.: An application of multiple Fourier series to identification of multivariable nonstationary systems. International Journal of Systems Science 20(10), 1993–2002 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Rutkowski, L.: Nonparametric learning algorithms in the time-varying environments. Signal Processing 18, 129–137 (1989)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rutkowski, L., Rafajłowicz, E.: On global rate of convergence of some nonparametric identification procedures. IEEE Transaction on Automatic Control AC-34(10), 1089–1091 (1989)CrossRefGoogle Scholar
  28. 28.
    Rutkowski, L.: Identification of MISO nonlinear regressions in the presence of a wide class of disturbances. IEEE Transactions on Information Theory IT-37, 214–216 (1991)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rutkowski, L.: Multiple Fourier series procedures for extraction of nonlinear regressions from noisy data. IEEE Transactions on Signal Processing 41(10), 3062–3065 (1993)zbMATHCrossRefGoogle Scholar
  30. 30.
    Rutkowski, L., Gałkowski, T.: On pattern classification and system identification by probabilistic neural networks. Applied Mathematics and Computer Science 4(3), 413–422 (1994)Google Scholar
  31. 31.
    Rutkowski, L.: A New Method for System Modelling and Pattern Classification. Bulletin of the Polish Academy of Sciences 52(1), 11–24 (2004)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Rutkowski, L., Cpałka, K.: A general approach to neuro - fuzzy systems. In: Proceedings of the 10th IEEE International Conference on Fuzzy Systems, Melbourne, December 2-5, vol. 3, pp. 1428–1431 (2001)Google Scholar
  33. 33.
    Rutkowski, L., Cpałka, K.: A neuro-fuzzy controller with a compromise fuzzy reasoning. Control and Cybernetics 31(2), 297–308 (2002)zbMATHGoogle Scholar
  34. 34.
    Scherer, R.: Boosting Ensemble of Relational Neuro-fuzzy Systems. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Żurada, J.M. (eds.) ICAISC 2006. LNCS (LNAI), vol. 4029, pp. 306–313. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  35. 35.
    Specht, D.F.: Probabilistic neural networks. Neural Networks 3, 109–118 (1990)CrossRefGoogle Scholar
  36. 36.
    Starczewski, J., Rutkowski, L.: Interval type 2 neuro-fuzzy systems based on interval consequents. In: Rutkowski, L., Kacprzyk, J. (eds.) Neural Networks and Soft Computing, pp. 570–577. Physica-Verlag, Springer-Verlag Company, Heidelberg, New York (2003)Google Scholar
  37. 37.
    Starczewski, J., Rutkowski, L.: Connectionist Structures of Type 2 Fuzzy Inference Systems. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2001. LNCS, vol. 2328, pp. 634–642. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  38. 38.
    Wilks, S.S.: Mathematical Statistics. John Wiley, New York (1962)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jacek M. Zurada
    • 1
  • Maciej Jaworski
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of LouisvilleLouisvilleUSA
  2. 2.Department of Computer EngineeringCzestochowa University of TechnologyCzestochowaPoland

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