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Study on semiparametric Wilcoxon fuzzy neural networks

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Abstract

Fuzzy neural network (FNN) has long been recognized as an efficient and powerful learning machine for general machine learning problems. Recently, Wilcoxon fuzzy neural network (WFNN), which generalizes the rank-based Wilcoxon approach for linear parametric regression problems to nonparametric neural network, was proposed aiming at improving robustness against outliers. FNN and WFNN are nonparametric models in the sense that they put no restrictions, except possibly smoothness, on the functional form of the regression function. However, they may be difficult to interpret and, even worse, yield poor estimates with high computational cost when the number of predictor variables is large. To overcome this drawback, semiparametric models have been proposed in statistical regression theory. A semiparametric model keeps the easy interpretability of its parametric part and retains the flexibility of its nonparametric part. Based on this, semiparametric FNN and semiparametric WFNN will be proposed in this paper. The learning rules are based on the backfitting procedure frequently used in semiparametric regression. Simulation results show that the semiparametric models perform better than their nonparametric counterparts.

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Abbreviations

\( \underline{m} \) :

\( \underline{m} : = \left\{ {1,\;2,\; \ldots ,\;m} \right\} \)

\( \Re^{n} \) :

Real n-space

med:

Median

A T :

Transpose of matrix A

X × Y :

Cartesian product of sets X and Y

[a, b]:

Interval between a and b

References

  • Angelov P, Xydeas C (2006) Fuzzy systems design: direct and indirect approaches. Soft Comput 10(9):836–849

    Article  Google Scholar 

  • Boser BE, Guyon IM, Vapnik VN (1992) A training algorithm for optimal margin classifiers. In: Haussler D (ed) Proceedings of the fifth annual workshop on computational learning theory. ACM Press, New York, pp 144–152

  • Breiman L, Friedman JH (1985) Estimating optimal transformations for multiple regression and correlations. J Am Stat Assoc 80:580–619

    Article  MATH  MathSciNet  Google Scholar 

  • Buja A, Hastie TJ, Tibshirani RJ (1989) Linear smoothers and additive models. Ann Stat 17:453–510

    Article  MATH  MathSciNet  Google Scholar 

  • Chuang CC, Su SF, Chen SS (2001) Robust TSK fuzzy modeling for function approximation with outliers. IEEE Trans Fuzzy Syst 9:810–821

    Article  Google Scholar 

  • Cortes C, Vapnik VN (1995) Support vector networks. Mach Learn 20:273–297

    MATH  Google Scholar 

  • Härdle W, Müller M, Sperlich S, Werwatz A (2004) Nonparametric and Semiparametric Models. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hartman EJ, Keeler JD, Kowalski JM (1990) Layered neural networks with Gaussian hidden units as universal approximations. Neural Comput 2:210–215

    Article  Google Scholar 

  • Hogg RV, McKean JW, Craig AT (2005) Introduction to mathematical statistics, 6th edn. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2(5):359–366

    Article  Google Scholar 

  • Hsieh JG, Lin YL, Jeng JH (2008) Preliminary study on Wilcoxon learning machines. IEEE Trans Neural Netw 19(2):201–211

    Article  Google Scholar 

  • Hung WL, Yang MS (2006) An omission approach for detecting outliers in fuzzy regression models. Fuzzy Sets Syst 157:3109–3122

    Article  MATH  MathSciNet  Google Scholar 

  • Kecman V (2001) Learning and soft computing. MIT Press, Cambridge

    MATH  Google Scholar 

  • Li H, Liu P (2005) Approximation analysis of feedforward regular fuzzy neural network with two hidden layers. Fuzzy Sets Syst 150(2):373–396

    Article  MATH  Google Scholar 

  • Liu P (2000) Analyses of regular fuzzy neural networks for approximation capabilities. Fuzzy Sets Syst 114:329–338

    Article  MATH  Google Scholar 

  • Liu P (2001) Universal approximations of continuous fuzzy-valued functions by multi-layer regular fuzzy neural networks. Fuzzy Sets Syst 119:313–320

    Article  MATH  Google Scholar 

  • Maindonald J, Braun J (2007) Data analysis and graphics using R, 2nd edn. Cambridge University Press, New York

    MATH  Google Scholar 

  • Otadi M, Mosleh M, Abbasbandy S (2011) Numerical solution of fully fuzzy linear systems by fuzzy neural network. Soft Comput (to appear)

  • Park J, Sandberg IW (1991) Universal approximation using radial basis function networks. Neural Comput 3:246–257

    Article  Google Scholar 

  • Rumelhart DE, Hinton GE, Williams RJ (1986) Learning internal representations by error propagation. In: Rumelhart DE, McClelland JL (eds) Parallel distributed processing: explorations in the microstructure of cognition, foundations, vol 1. MIT Press, Cambridge, pp 318–362

  • Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  • Tsai HH, Yu PT (2000) On the optimal design of fuzzy neural networks with robust learning for function approximation. IEEE Trans Syst Man Cybern B Cybern 30:217–223

    Article  Google Scholar 

  • Wang LX (1997) A Course in Fuzzy Systems and Control. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Wang WY, Lee TT, Liu CL, Wang CH (1997) Function approximation using fuzzy neural networks with robust learning algorithm. IEEE Trans Syst Man Cybern B Cybern 27(4):740–747

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan

    Hsu-Kun Wu

  2. Department of Information Engineering, I-Shou University, Kaohsiung, 84001, Taiwan

    Yih-Lon Lin & Jyh-Horng Jeng

  3. Department of Electrical Engineering, I-Shou University, Kaohsiung, 84001, Taiwan

    Jer-Guang Hsieh

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  1. Hsu-Kun Wu
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  2. Yih-Lon Lin
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  3. Jer-Guang Hsieh
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  4. Jyh-Horng Jeng
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Correspondence to Yih-Lon Lin.

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Wu, HK., Lin, YL., Hsieh, JG. et al. Study on semiparametric Wilcoxon fuzzy neural networks. Soft Comput 16, 11–21 (2012). https://doi.org/10.1007/s00500-011-0730-3

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