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Nonparametric Estimation of Edge Values of Regression Functions

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Artificial Intelligence and Soft Computing (ICAISC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9693))

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Abstract

In this article we investigate the problem of regression functions estimation in the edges points of their domain. We refer to 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, \(x_i \in D\), \(y_i\) is the set of probabilistic outputs, and \(\epsilon _i\) is a measurement noise with zero mean and bounded variance. R(.) is a completely unknown function. The possible solution of finding unknown function is to apply the algorithms based on the Parzen kernel [13, 31]. The commonly known drawback of these algorithms is that the error of estimation dramatically increases if the point of estimation x is drifting to the left or right bound of interval D. This fact makes it impossible to estimate functions exactly in edge values of domain.

The main goal of this paper is an application of NMS algorithm (introduced in [11]), basing on integral version of the Parzen method of function estimation by combining the linear approximation idea. The results of numerical experiments are presented.

M. Pawlak carried out this research at ASS during his sabbatical leave from University of Manitoba.

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References

  1. Aghdam, M.H., Heidari, S.: Feature selection using particle swarm optimization in text categorization. J. Artif. Intell. Soft Comput. Res. 5(4), 231–238 (2015)

    Article  Google Scholar 

  2. Bas, E.: The Training of multiplicative neuron model artificial neural networks with differential evolution algorithm for forecasting. J. Artif. Intell. Soft Comput. Res. 6(1), 5–11 (2016)

    Article  Google Scholar 

  3. Bertini Jr., J.R., Carmo, N.M.: Enhancing constructive neural network performance using functionally expanded input data. J. Artif. Intell. Soft Comput. Res. 6(2), 119–131 (2016)

    Google Scholar 

  4. Chen, S.X.: Beta kernel estimators for density functions. J. Stat. Plann. Infer. 139, 2269–2283 (2009)

    Article  Google Scholar 

  5. Chu, J.L., Krzyzak, A.: The recognition of partially occluded objects with support vector machines, convolutional neural networks and deep belief networks. J. Artif. Intell. Soft Comput. Res. 4(1), 5–19 (2014)

    Article  Google Scholar 

  6. Cierniak, R., Rutkowski, L.: On image compression by competitive neural networks and optimal linear predictors. Sig. Process.-Image Commun. 15(6), 559–565 (2000)

    Article  Google Scholar 

  7. Duch, W., Korbicz, J., Rutkowski, L., Tadeusiewicz, R. (eds.): Biocybernetics and Biomedical Engineering 2000. Neural Networks, vol. 6. Akademicka Oficyna Wydawnicza, EXIT, Warsaw (2000) (in Polish)

    Google Scholar 

  8. Galkowski, T., Rutkowski, L.: Nonparametric recovery of multivariate functions with applications to system identification. In: Proceedings of the IEEE, vol. 73, pp. 942–943, New York (1985)

    Google Scholar 

  9. Galkowski, T., Rutkowski, L.: Nonparametric fitting of multivariable functions. IEEE Trans. Autom. Control AC–31, 785–787 (1986)

    Article  MATH  Google Scholar 

  10. Galkowski, T.: Nonparametric estimation of boundary values of functions. Arch. Control Sci. 3(1–2), 85–93 (1994)

    MathSciNet  MATH  Google Scholar 

  11. 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, vol. 7895, pp. 158–166. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  12. Galkowski, T., Pawlak, M.: Nonparametric extension of regression functions outside domain. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2014, Part I. LNCS, vol. 8467, pp. 518–530. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

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

    MATH  Google Scholar 

  14. Greblicki, W., Rutkowski, L.: Density-free bayes risk consistency of nonparametric pattern recognition procedures. Proc. IEEE 69(4), 482–483 (1981)

    Article  Google Scholar 

  15. Greblicki, W., Rutkowska, D., Rutkowski, L.: An orthogonal series estimate of time-varying regression. Ann. Inst. Stat. Math. 35(1), 215–228 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hazelton, M.L., Marshall, J.C.: Linear boundary kernels for bivariate density estimation. Stat. Prob. Lett. 79, 999–1003 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Karunamuni, R.J., Alberts, T.: On boundary correction in kernel density estimation. Stat. Methodol. 2, 191–212 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Karunamuni, R.J., Alberts, T.: A locally adaptive transformation method of boundary correction in kernel density estimation. J. Stat. Plann. Infer. 136, 2936–2960 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kitajima, R., Kamimura, R.: Accumulative information enhancement in the self-organizing maps and its application to the analysis of mission statements. J. Artif. Intell. Soft Comput. Res. 5(3), 161–176 (2015)

    Article  Google Scholar 

  20. Knop, M., Kapuscinski, T., Mleczko, W.K.: Video key frame detection based on the restricted Boltzmann machine. J. Appl. Math. Comput. Mech. 14(3), 49–58 (2015)

    Article  Google Scholar 

  21. Korytkowski, M., Nowicki, R., Scherer, R.: Neuro-fuzzy rough classifier ensemble. In: Alippi, C., Polycarpou, M., Panayiotou, C., Ellinas, G. (eds.) ICANN 2009, Part I. LNCS, vol. 5768, pp. 817–823. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  22. Korytkowski, M., Rutkowski, L., Scherer, R.: Fast image classification by boosting fuzzy classifiers. Inf. Sci. 327, 175–182 (2016)

    Article  MathSciNet  Google Scholar 

  23. Koshiyama, A.S., Vellasco, M., Tanscheit, R.: GPFIS-control: a genetic fuzzy system for control tasks. J. Artif. Intell. Soft Comput. Res. 4(3), 167–179 (2014)

    Article  Google Scholar 

  24. Kyung-Joon, C., Schucany, W.R.: Nonparametric kernel regression estimation near endpoints. J. Stat. Plann. Inf. 66, 289–304 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Marshall, J.C., Hazelton, M.L.: Boundary kernels for adaptive density estimators on regions with irregular boundaries. J. Multivar. Anal. 101, 949–963 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Laskowski, L.: A novel hybrid-maximum neural network in stereo-matching process. Neural Comput. Appl. 23(7–8), 2435–2450 (2013)

    Article  Google Scholar 

  27. Laskowski, L., Jelonkiewicz, J.: Self-correcting neural network for stereo-matching problem solving. Fundamenta Informaticae 138, 1–26 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Nikulin, V.: Prediction of the shoppers loyalty with aggregated data streams. J. Artif. Intell. Soft Comput. Res. 6(2), 69–79 (2016)

    Article  MathSciNet  Google Scholar 

  30. Nowak, B.A., Nowicki, R.K., Starczewski, J.T., Marvuglia, A.: The learning of neuro-fuzzy classifier with fuzzy rough sets for imprecise datasets. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2014, Part I. LNCS, vol. 8467, pp. 256–266. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  31. Parzen, E.: On estimation of a probability density function and mode. Anal. Math. Stat. 33(3), 1065–1076 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  32. Poměnková-Dluhá, J.: Edge Effects of Gasser-Müller Estimator, Mathematica 15, Brno, Masaryk University, pp. 307–314 (2004)

    Google Scholar 

  33. Rafajlowicz, E.: Nonparametric least squares estimation of a regression function statistics. J. Theor. Appl. Stat. 19(3), 349–358 (1988)

    MathSciNet  MATH  Google Scholar 

  34. Rafajlowicz, E., Schwabe, R.: Halton and hammersley sequences in multivariate nonparametric regression. Stat. Prob. Lett. 76(8), 803–812. Elsevier (2006)

    Google Scholar 

  35. Rutkowska, A.: Influence of membership function’s shape on portfolio optimization results. J. Artif. Intell. Soft Comput. Res. 6(1), 45–54 (2016)

    Article  Google Scholar 

  36. Rutkowski, L.: Sequential estimates of probability densities by orthogonal series and their application in pattern classification. IEEE Trans. Syst. Man Cybern. SMC–10(12), 918–920 (1980)

    MathSciNet  MATH  Google Scholar 

  37. Rutkowski, L.: On bayes risk consistent pattern recognition procedures in a quasi-stationary environment. IEEE Trans. Pattern Anal. Mach. Intell. PAMI–4(1), 84–87 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rutkowski, L.: Online identification of time-varying systems by nonparametric techniques. IEEE Trans. Autom. Control 27(1), 228–230 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rutkowski, L.: On nonparametric identification with prediction of time-varying systems. IEEE Trans. Autom. Control AC–29, 58–60 (1984)

    Article  MATH  Google Scholar 

  40. Rutkowski, L.: Nonparametric identification of quasi-stationary systems. Syst. Control Lett. 6, 33–35. Amsterdam (1985)

    Google Scholar 

  41. Rutkowski, L.: Real-time identification of time-varying systems by non-parametric algorithms based on parzen kernels. Int. J. Syst. Sci. 16, 1123–1130 (1985)

    Article  MATH  Google Scholar 

  42. Rutkowski, L.: A general approach for nonparametric fitting of functions and their derivatives with applications to linear circuits identification. IEEE Trans. Circuits Syst. 33, 812–818 (1986)

    Article  MATH  Google Scholar 

  43. Rutkowski, L.: Sequential pattern recognition procedures derived from multiple Fourier series. Pattern Recogn. Lett. 8, 213–216 (1988)

    Article  MATH  Google Scholar 

  44. Rutkowski, L.: Application of multiple fourier series to identification of multivariable nonstationary systems. Int. J. Syst. Sci. 20(10), 1993–2002 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rutkowski, L.: Non-parametric learning algorithms in the time-varying environments. Sig. Process. 18(2), 129–137 (1989)

    Article  MathSciNet  Google Scholar 

  46. Rutkowski, L., Rafajłowicz, E.: On global rate of convergence of some nonparametric identification procedures. IEEE Trans. Autom. Control AC–34(10), 1089–1091 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rutkowski, L.: Identification of MISO nonlinear regressions in the presence of a wide class of disturbances. IEEE Trans. Inf. Theory IT–37, 214–216 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  48. Rutkowski, L.: Multiple fourier series procedures for extraction of nonlinear regressions from noisy data. IEEE Trans. Sig. Process. 41(10), 3062–3065 (1993)

    Article  MATH  Google Scholar 

  49. Rutkowski, L.: Generalized regression neural networks in time-varying environment. IEEE Trans. Neural Netw. 15(3), 576–596 (2004)

    Article  Google Scholar 

  50. Rutkowski, L.: Adaptive probabilistic neural networks for pattern classification in time-varying environment. IEEE Trans. Neural Netw. 15(4), 811–827 (2004)

    Article  MathSciNet  Google Scholar 

  51. Rutkowski, L., Pietruczuk, L., Duda, P., Jaworski, M.: Decision trees for mining data streams based on the mcdiarmid’s bound. IEEE Trans. Knowl. Data Eng. 25(6), 1272–1279 (2013)

    Article  Google Scholar 

  52. Rutkowski, L., Jaworski, M., Duda, P., Pietruczuk, L.: Decision trees for mining data streams based on the gaussian approximation. IEEE Trans. Knowl. Data Eng. 26(1), 108–119 (2014)

    Article  Google Scholar 

  53. Rutkowski, L., Jaworski, M., Pietruczuk, L., Duda, P.: The CART decision trees mining data streams. Inf. Sci. 266, 1–15 (2014)

    Article  Google Scholar 

  54. Rutkowski, L., Jaworski, M., Pietruczuk, L., Duda, P.: A new method for data stream mining based on the misclassification error. IEEE Trans. Neural Netw. Learn. Syst. 26(5), 1048–1059 (2015)

    Article  MathSciNet  Google Scholar 

  55. Schuster, E.F.: Incorporating support constraints into nonparametric estimators of densities. Commun. Stat. Part A - Theory Methods 14, 1123–1136 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  56. Skubalska-Rafajlowicz, E.: Pattern recognition algorithms based on space-filling curves and orthogonal expansions. IEEE Trans. Inf. Theory 47(5), 1915–1927 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  57. Skubalska-Rafajlowicz, E.: Random projection RBF nets for multidimensional density estimation. Int. J. Appl. Math. Comput. Sci. 18(4), 455–464 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  58. Szarek, A., Korytkowski, M., Rutkowski, L., Scherer, R., Szyprowski, J.: Application of neural networks in assessing changes around implant after total hip arthroplasty. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2012, Part II. LNCS, vol. 7268, pp. 335–340. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  59. Wang, Z., Zhang-Westmant, L.: New ranking method for fuzzy numbers by their expansion center. J. Artif. Intell. Soft Comput. Res. 4(3), 181–187 (2014)

    Article  Google Scholar 

  60. Zhang, S., Karunamuni, R.J.: On kernel density estimation near endpoints. J. Stat. Plann. Inf. 70, 301–316 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhang, S., Karunamuni, R.J.: Deconvolution boundary kernel method in nonparametric density estimation. J. Stat. Plann. Inf. 139, 2269–2283 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhang, S., Karunamuni, R.J.: Boundary performance of the beta kernel estimators. Nonparametric Stat. 22, 81–104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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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
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  2. Miroslaw Pawlak
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Corresponding author

Correspondence to Tomasz Galkowski .

Editor information

Editors and Affiliations

  1. Częstochowa University of Technology, Czestochowa, Poland

    Leszek Rutkowski

  2. Częstochowa University of Technology, Czestochowa, Poland

    Marcin Korytkowski

  3. Częstochowa University of Technology, Czestochowa, Poland

    Rafał Scherer

  4. AGH University of Science and Technology, Krakow, Poland

    Ryszard Tadeusiewicz

  5. University of California, Berkeley, California, USA

    Lotfi A. Zadeh

  6. University of Louisville, Louisville, Kentucky, USA

    Jacek M. Zurada

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Galkowski, T., Pawlak, M. (2016). Nonparametric Estimation of Edge Values of Regression Functions. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L., Zurada, J. (eds) Artificial Intelligence and Soft Computing. ICAISC 2016. Lecture Notes in Computer Science(), vol 9693. Springer, Cham. https://doi.org/10.1007/978-3-319-39384-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-39384-1_5

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