Skip to main content
SpringerLink
Log in
Book cover

International Conference on Artificial Intelligence and Soft Computing

ICAISC 2014: Artificial Intelligence and Soft Computing pp 518–530Cite as

Nonparametric Extension of Regression Functions Outside Domain

  • Conference paper

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

Abstract

The article refers to the problem of regression functions estimation in the points situated near the edges but outside of function domain. We investigate the model \(y_i = R\left( {x_i } \right) + \epsilon _i ,\,i = 1,2, \ldots n\), where x i is assumed to be the set of deterministic inputs, x i  ∈ D, y i is the set of probabilistic outputs, and ε i is a measurement noise with zero mean and bounded variance. R(.) is a completely unknown function. In the literature the possible ways of finding unknown function are based on the algorithms derived from the Parzen kernel. These algorithms were also applied to estimation of the derivatives of unknown functions. The commonly known disadvantage of the kernel algorithms is that the error of estimation dramatically increases if the point of estimation x is approaching to the left or right bound of interval D. Algorithms on predicting values in the boundary region outside the function domain D are unknown for the author, so far.

The main result of this paper is a new algorithm based on integral version of Parzen methods for local prediction of values of the function R near boundaries in the region outside domain. The results of numerical experiments are presented.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chen, S.X.: Beta kernel estimators for density functions. Journal of Statistical Planning and Inference 139, 2269–2283 (2009)

    Article  MathSciNet  Google Scholar 

  2. 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 

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

    Article  Google Scholar 

  4. Galkowski, T.: Nonparametric estimates of parameters in a model of microbial growth process. AMSE Review 8(1), 1–10 (1988)

    Google Scholar 

  5. Galkowski, T.: On nonparametric fitting of higher order function derivatives by the kernel method - a simulation study. In: Gutierrez, R., Valderrama, M.J. (eds.) Fifth Int. Symp. Applied Stochastic Models and Data Analysis, Granada, Spain, pp. 230–242 (1991)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  7. 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 

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

    Google Scholar 

  9. Gasser, T., Muller, H.G.: Estimating Regression Functions and Their Derivatives by the Kernel Method. Scandinavian Journal of Statistics 11(3), 171–185 (1984)

    MATH  MathSciNet  Google Scholar 

  10. Grbic, R., Kurtagic, D., Sliškovic, D.: Stream water temperature prediction based on Gaussian process regression. Expert Systems with Applications 40, 7407–7414 (2013)

    Article  Google Scholar 

  11. 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 

  12. 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 

  13. Hazelton, M.L., Marshall, J.C.: Linear boundary kernels for bivariate density estimation. Statistics and Probability Letters 79, 999–1003 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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 (LNAI), vol. 7268, pp. 82–91. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  15. Karunamuni, R.J., Alberts, T.: On boundary correction in kernel density estimation. Statistical Methodology 2, 191–212 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Karunamuni, R.J., Alberts, T.: A locally adaptive transformation method of boundary correction in kernel density estimation. Journal of Statistical Planning and Inference 136, 2936–2960 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. 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, vols. 1-10, pp. 1274–1277 (2006)

    Google Scholar 

  18. 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 

  19. Kyung-Joon, C., Schucany, W.R.: Nonparametric kernel regression estimation near endpoints. Journal of Statistical Planning and Inference 66, 289–304 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Landberg, L.: Short-term prediction of local wind conditions. Journal of Wind Engineering and Industrial Aerodynamics 89, 235–245 (2001)

    Article  Google Scholar 

  21. Marshall, J.C., Hazelton, M.L.: Boundary kernels for adaptive density estimators on regions with irregular boundaries. Journal of Multivariate Analysis 101, 949–963 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Min, W., Wynter, L.: Real-time road traffic prediction with spatio-temporal correlations. Transportation Research Part C 19, 606–616 (2011)

    Article  Google Scholar 

  23. Mosallam, A., Medjaher, K., Zerhouni, N.: Nonparametric time series modelling for industrial prognostics and health management. Int. J. Adv. Manuf. Technol. 69, 1685–1699 (2013)

    Article  Google Scholar 

  24. Müller, H.G.: Smooth optimum kernel estimators near endpoints. Biometrika 78, 521–530 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  25. Prampero Paulo, S., Romis, A.: Magnetic particle swarm optimization. Journal of Artificial Intelligence and Soft Computing Research 2(1), 59–72 (2012)

    Google Scholar 

  26. 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 

  27. 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 

  28. 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 

  29. Poměnková-Dluhá, J.: Edge Effects of Gasser-Müller Estimator. Mathematica 15, 307–314 (2004)

    Google Scholar 

  30. 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 

  31. 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 

  32. 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 

  33. 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 

  34. 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 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. 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 

  38. 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)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  40. 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 

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

    Article  MathSciNet  Google Scholar 

  42. 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 

  43. 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 

  44. 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 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  47. 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 

  48. 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 

  49. 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 

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

    Google Scholar 

  51. 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 

  52. 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 

  53. Schuster, E.F.: Incorporating Support Constraints Into Nonparametric Estimators of Densities. Communications in Statistics, Part A - Theory and Methods 14, 1123–1136 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  54. Strang, G.: Calculus. MIT book, Wellesley-Cambridge Press (1991)

    Google Scholar 

  55. Vu Nam, T., Brdys Mietek, A.: Optimizing control by robustly feasible model predictive control and application to drinking water distribution systems. Journal of Artificial Intelligence and Soft Computing Research 1(1), 43–57 (2011)

    Google Scholar 

  56. Vilar, J.M., Cao, R., Aneiros, G.: Forecasting next-day electricity demand and price using nonparametric functional methods. Electrical Power and Energy Systems 39, 48–55 (2012)

    Article  Google Scholar 

  57. Zhang, J., Tan, Z.: Day-ahead electricity price forecasting using WT, CLSSVM and EGARCH model. Electrical Power and Energy Systems 45, 362–368 (2013)

    Article  MathSciNet  Google Scholar 

  58. Zhang, S., Karunamuni, R.J.: On kernel density estimation near endpoints. Journal of Statistical Planning and Inference 70, 301–316 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  59. Zhang, S., Karunamuni, R.J.: Deconvolution boundary kernel method in nonparametric density estimation. Journal of Statistical Planning and Inference 139, 2269–2283 (2009)

    Article  MATH  MathSciNet  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 Extension of Regression Functions Outside Domain. 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_44

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-07173-2_44

  • 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