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Semiparametric Inference in Identification of Block-Oriented Systems

Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 605)

Abstract

In this paper, we give the semiparametric statistics perspective on the problem of identification of a class of nonlinear dynamic systems. We present a framework for identification of the so-called block-oriented systems that can be represented by finite-dimensional parameters and an infinite-dimensional set of nonlinear characteristics that run typically through a nonparametric class of univariate functions. We consider systems which are expressible exactly in this form and the case when they are approximative models. In the latter case, we derive projections, that is, solutions which minimize the mean \(L_2\) error. The chief benefit of such an approach is to make classical nonparametric estimates amenable to the incorporation of constraints and able to overcome the celebrated curse of dimensionality and system complexity. The developed methodology is explained by examining semiparametric versions of popular block-oriented structures, i.e., Hammerstein, Wiener, and parallel systems.

Notes

Acknowledgments

The author wishes to thank Mount First for assistance. We would also like to thank the referee for his valuable comments and corrections.

References

  1. 1.
    Andrews DWK (1984) Non-strong mixing autoregressive process. J Appl Probab 21:930–934MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Billings S (2013) Nonlinear system identification. Wiley, New YorkGoogle Scholar
  3. 3.
    Boyd S, Chua L (1985) Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans Circuits Syst 32:1150–1161MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cohen A, Daubechies I, DeVore R, Kerkyacharian G, Picard D (2012) Capturing ridge functions in high dimensions from point queries. Contr Approx 35:225–243MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Devroye L (1988) Automatic pattern recognition: a study of the probability of error. IEEE Trans Pattern Anal Mach Intell 10:530–543Google Scholar
  6. 6.
    Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recogntion. Springer, New YorkGoogle Scholar
  7. 7.
    Diaconis P, Shahshahani M (1984) On nonlinear functions of linear combinations. SIAM J Sci Comput 5(1):175–191MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Espinozo M, Suyken JAK, De Moor B (2005) Kernel based partially linear models and nonlinear identification. IEEE Trans Autom Contr 50:1602–1606CrossRefGoogle Scholar
  9. 9.
    Fan J, Yao Q (2003) Nonlinear time series: nonparametric and parametric methods. Springer, New YorkGoogle Scholar
  10. 10.
    Giannakis GB, Serpendin E (2001) A bibliography on nonlinear system identification. Sig Process 81:533–580CrossRefMATHGoogle Scholar
  11. 11.
    Giri F, Bai EW (eds) (2010) Block-oriented nonlinear system identification. Springer, New YorkGoogle Scholar
  12. 12.
    Greblicki W (2010) Nonparametric input density-free estimation of nonlinearity in Wiener systems. IEEE Trans Inform Theory 56:3575–3580MathSciNetCrossRefGoogle Scholar
  13. 13.
    Greblicki W, Pawlak M (2008) Nonparametric system identification. Cambridge University Press, CambridgeGoogle Scholar
  14. 14.
    Härdle W, Hall P, Ichimura H (1993) Optimal smoothing in single-index models. Ann Stat 21:157–178CrossRefMATHGoogle Scholar
  15. 15.
    Härdle W, Müller M, Sperlich S, Werwatz A (2004) Nonparametric and semiparametric models. Springer, New YorkGoogle Scholar
  16. 16.
    Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning. Springer, New YorkGoogle Scholar
  17. 17.
    Isermann R, Munchhof M (2011) Identification of dynamic systems: an introduction with applications. Springer, New YorkGoogle Scholar
  18. 18.
    Koronacki J, Ćwik J (2008) Statystyczne systemy uczace sie (in Polish). Exit, WarsawGoogle Scholar
  19. 19.
    Kvam PH, Vidakovic B (2007) Nonparametric statistics with applications to science and engineering. Wiley, New YorkGoogle Scholar
  20. 20.
    Li Q, Racine JS (2007) Nonparametric econometrics. Princeton University Press, PrincetonGoogle Scholar
  21. 21.
    Ljung L (1999) System identification: theory for the user. Prentice-Hall, Englewood CliffsGoogle Scholar
  22. 22.
    Meyn SP, Tweedie RL (1993) Markov chain and stochastic stability. Springer, New YorkGoogle Scholar
  23. 23.
    Mohri M, Rostamizadeh A, Talwalker A (2012) Foundations of machine learning. The MIT Press, CambridgeGoogle Scholar
  24. 24.
    Pawlak M, Lv J (2014) On nonparametric identification of MISO Hammerstein systemsGoogle Scholar
  25. 25.
    Pawlak M, Hasiewicz Z, Wachel P (2007) On nonparametric identification of Wiener systems. IEEE Trans Signal Process 55:482–492MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pillonetto G, Dinuzzo F, Che T, De Nicolao G, Ljung L (2014) Kernel methods in system identification, machine learning and function estimation: a survey. Automatica 50:657–682CrossRefMATHGoogle Scholar
  27. 27.
    Saart P, Gao J, Kim NH (2014) Semiparametric methods in nonlinear time series: a selective review. J Nonparametric Stat 26:141–169MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, CambridgeGoogle Scholar
  29. 29.
    Vidyasagar M, Karandikar RL (2008) A learning theory approach to system identification and stochastic adaptive control. J Process Contr 18:421–430CrossRefGoogle Scholar
  30. 30.
    Wasserman L (2006) All of nonparametric statistics. Springer, New YorkGoogle Scholar
  31. 31.
    Westwick D, Kearney R (2003) Identification of nonlinear physiological systems. Wiley, New YorkGoogle Scholar
  32. 32.
    Wu WB (2005) Nonlinear system theory: another look at dependence. Proc Nat Acad Sci 102:14150–14154CrossRefMATHGoogle Scholar
  33. 33.
    Wu WB, Mielniczuk J (2010) A new look at measuring dependence. In: Doukham P et al. (eds) Dependence in probability and statistics. Springer, New York, pp 123–142Google Scholar
  34. 34.
    Yatchev A (2003) Semiparametric regression for the applied econometrician. Cambridge University Press, CambridgeGoogle Scholar
  35. 35.
    Yatracos Y (1989) On the estimation of the derivative of a function with the derivative of an estimate. J Multivar Anal 28:172–175MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of ManitobaWinnipegCanada

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