Skip to main content
SpringerLink
Log in

Nonparametric Function Fitting in the Presence of Nonstationary Noise

  • Conference paper
Artificial Intelligence and Soft Computing (ICAISC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8467))

Included in the following conference series:

Abstract

The article refers to the problem of regression functions estimation in the presence of nonstationary noise. We investigate the model \(y_i = R\left( {{\bf x _i}} \right) + \epsilon _i ,\,i = 1,2, \ldots n\), where x i is assumed to be the d-dimensional vector, set of deterministic inputs, x i  ∈ S d, y i is the scalar, set of probabilistic outputs, and ε i is a measurement noise with zero mean and variance depending on n. \(R\left( . \right)\) is a completely unknown function. One of the possible solutions of finding function \(R\left( . \right)\) is to apply non-parametric methodology - algorithms based on the Parzen kernel or algorithms derived from orthogonal series. The novel result of this article is the analysis of convergence for some class of nonstationarity. We present the conditions when the algorithm of estimation is convergent even when the variance of noise is divergent with number of observations tending to infinity. The results of numerical experiments are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Galkowski, T., Rutkowski, L.: Nonparametric recovery of multivariate functions with applications to system identification. Proceedings of the IEEE 73, 942–943 (1985)

    Article  Google Scholar 

  2. Galkowski, T., Rutkowski, L.: Nonparametric fitting of multivariable functions. IEEE Transactions on Automatic Control AC-31, 785–787 (1986)

    Article  Google Scholar 

  3. Galkowski, T.: Nonparametric estimation of boundary values of functions. Archives of Control Science 3(1-2), 85–93 (1994)

    MATH  MathSciNet  Google Scholar 

  4. Gałkowski, T.: Kernel estimation of regression functions in the boundary regions. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part II. LNCS (LNAI), vol. 7895, pp. 158–166. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  5. Gasser, T., Muller, H.G.: Kernel estimation of regression functions. Lecture Notes in Mathematics, vol. 757, pp. 23–68. Springer, Heidelberg (1979)

    Google Scholar 

  6. Greblicki, W., Rutkowski, L.: Density-free Bayes risk consistency of nonparametric pattern recognition procedures. Proceedings of the IEEE 69(4), 482–483 (1981)

    Article  Google Scholar 

  7. Greblicki, W., Rutkowska, D., Rutkowski, L.: An orthogonal series estimate of time-varying regression. Annals of the Institute of Statistical Mathematics 35(2), 215–228 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. Jaworski, M., Duda, P., Pietruczuk, L.: On fuzzy clustering of data streams with concept drift. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2012, Part II. LNCS, vol. 7268, pp. 82–91. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  9. Korytkowski, M., Rutkowski, L., Scherer, R.: On combining backpropagation with boosting. In: IEEE International Joint Conference on Neural Network (IJCNN) Proceedings, Vancouver, July 16-21, vol. 1-10, pp. 1274–1277 (2006)

    Google Scholar 

  10. Korytkowski, M., Rutkowski, L., Scherer, R.: From Ensemble of Fuzzy Classifiers to Single Fuzzy Rule Base Classifier. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2008. LNCS (LNAI), vol. 5097, pp. 265–272. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  11. Kroll, A.: On choosing the fuzziness parameter for identifying TS models with multidimensional membership functions. Journal of Artificial Intelligence and Soft Computing Research 1(4), 283–300 (2011)

    Google Scholar 

  12. Lobato, F.S., Steffen Jr., V., Neto, A.J.S.: Solution of singular optimal control problems using the improved differential evolution algorithm. Journal of Artificial Intelligence and Soft Computing Research 1(3), 195–206 (2011)

    Google Scholar 

  13. Parzen, E.: On estimation of a probability density function and mode. Analysis of Mathematical Statistics 33(3), 1065–1076 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pietruczuk, L., Duda, P., Jaworski, M.: A new fuzzy classifier for data streams. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2012, Part I. LNCS (LNAI), vol. 7267, pp. 318–324. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  15. Pietruczuk, L., Duda, P., Jaworski, M.: Adaptation of decision trees for handling concept drift. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part I. LNCS (LNAI), vol. 7894, pp. 459–473. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  16. Rafajlowicz, E.: Nonparametric orthogonal series estimators of regression - A class attaining the optimal convergence in L2. Statistics and Probability Letters 5(3), 219–224 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rafajlowicz, E., Schwabe, R.: Halton and Hammersley sequences in multivariate nonparametric regression. Statistics and Probability Letters 76(8), 803–812 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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)

    MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Rutkowski, L.: On-line identification of time-varying systems by nonparametric techniques. IEEE Transactions on Automatic Control AC-27, 228–230 (1982)

    Article  Google Scholar 

  21. Rutkowski, L.: On nonparametric identification with prediction of time-varying systems. IEEE Transactions on Automatic Control AC-29, 58–60 (1984)

    Article  Google Scholar 

  22. Rutkowski, L.: Nonparametric identification of quasi-stationary systems. Systems and Control Letters 6, 33–35 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rutkowski, L.: The real-time identification of time-varying systems by non-parametric algorithms based on Parzen kernels. International Journal of Systems Science 16, 1123–1130 (1985)

    Article  MATH  Google Scholar 

  24. Rutkowski, L.: A general approach for nonparametric fitting of functions and their derivatives with applications to linear circuits identification. IEEE Transactions on Circuits Systems CAS-33, 812–818 (1986)

    Article  Google Scholar 

  25. Rutkowski, L.: Sequential pattern recognition procedures derived from multiple Fourier series. Pattern Recognition Letters 8, 213–216 (1988)

    Article  MATH  Google Scholar 

  26. Rutkowski, L.: An application of multiple Fourier series to identification of multivariable nonstationary systems. International Journal of Systems Science 20(10), 1993–2002 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rutkowski, L.: Non-parametric learning algorithms in the time-varying environments. Signal Processing 18(2), 129–137 (1989)

    Article  MathSciNet  Google Scholar 

  28. Rutkowski, L., Rafajlowicz, E.: On optimal global rate of convergence of some nonparametric identification procedures. IEEE Transaction on Automatic Control AC-34(10), 1089–1091 (1989)

    Article  MathSciNet  Google Scholar 

  29. 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)

    Article  MathSciNet  Google Scholar 

  30. Rutkowski, L.: Multiple Fourier series procedures for extraction of nonlinear regressions from noisy data. IEEE Transactions on Signal Processing 41(10), 3062–3065 (1993)

    Article  MATH  Google Scholar 

  31. Rutkowski, L.: Adaptive probabilistic neural-networks for pattern classification in time-varying environment. IEEE Trans. Neural Networks 15, 811–827 (2004)

    Article  Google Scholar 

  32. Rutkowski, L.: Generalized regression neural networks in time-varying environment. IEEE Trans. Neural Networks 15, 576–596 (2004)

    Article  Google Scholar 

  33. Rutkowski, L., Przybyl, A., Cpalka, K.: Novel on-line speed profile generation for industrial machine tool based on flexible neuro-fuzzy approximation. IEEE Transactions on Industrial Electronics 59, 1238–1247 (2012)

    Article  Google Scholar 

  34. Rutkowski, L., Pietruczuk, L., Duda, P., Jaworski, M.: Decision trees for mining data streams based on the McDiarmid’s bound. IEEE Transactions on Knowledge and Data Engineering 25(6), 1272–1279 (2013)

    Article  Google Scholar 

  35. Rutkowski, L., Jaworski, M., Pietruczuk, L., Duda, P.: Decision trees for mining data streams based on the gaussian approximation. IEEE Transactions on Knowledge and Data Engineering 26(1), 108–119 (2014)

    Article  Google Scholar 

  36. Rutkowski, L., Jaworski, M., Pietruczuk, L., Duda, P.: The CART decision tree for mining data streams. Information Sciences 266, 1–15 (2014)

    Google Scholar 

  37. 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 

  38. Starczewski, J.T., 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)

    Chapter  Google Scholar 

  39. Theodoridis, D.C., Boutalis, Y.S., Christodoulou, M.A.: Robustifying analysis of the direct adaptive control of unknown multivariable nonlinear systems based on a new neuro-fuzzy method. Journal of Artificial Intelligence and Soft Computing Research 1(1), 59–79 (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Intelligence, Czestochowa University of Technology, Czestochowa, Poland

    Tomasz Galkowski

  2. Information Technology Institute, University of Social Sciences, Lodz, Poland

    Miroslaw Pawlak

  3. Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Canada

    Miroslaw Pawlak

Authors
  1. Tomasz Galkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miroslaw Pawlak
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Częstochowa University of Technology, Armii Krajowej 36, 42-200, Częstochowa, Poland

    Leszek Rutkowski , Marcin Korytkowski  & Rafał Scherer ,  & 

  2. AGH University of Science and Technology, Mickiewicza 30, 30-059, Kraków, Poland

    Ryszard Tadeusiewicz

  3. Computer Science Division, Department of Electrical Engineering and Computer Sciences, University of California Berkeley, 94720-1776, Berkeley, CA, USA

    Lotfi A. Zadeh

  4. Computational Intelligence Laboratory, Electrical and Computer Engineering, University of Louisville, 405 Lutz Hall, 40292, Louisville, KY, USA

    Jacek M. Zurada

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Galkowski, T., Pawlak, M. (2014). Nonparametric Function Fitting in the Presence of Nonstationary Noise. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC 2014. Lecture Notes in Computer Science(), vol 8467. Springer, Cham. https://doi.org/10.1007/978-3-319-07173-2_45

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-07173-2_45

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07172-5

  • Online ISBN: 978-3-319-07173-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation