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On Kernel Estimation Using Non-Gaussian and/or Non-White Input Data

  • Vasilis Z. Marmarelis

Abstract

In many actual applications of the Volterra-Wiener approach, the prevailing conditions are less than ideal ie., the input-output data are contaminated by noise and/or the available input data deviate from the Gaussian white noise archetype required by Wiener’s theory. Deviations from whiteness and/orGaussianness may be ca used by experimental necessity or inadvertent stimulus distortion. At the same time, experimen tal and/or computational expediencies have led to the introduction of non-Gaussian multi-level (e.g., binary, ternary, etc.) or multi-frequency quasi-white (i.e., approximately white) input signals. Since the original (and most widely used to date) kernel estimation technique is based on input-output crosscorrelations and requires input whiteness (or, at least, quasi-whiteness), extensive efforts have been made to adapt this crosscorrelation technique to non-Gaussian (but quasi-white) inputs. These efforts have led to successful adaptations of the cross-correlation technique to several classes of non-Gaussian quasi-white inputs, although certain restrictions apply in some cases (e.g., binary white inputs cannot estimate diagonal kernel values). More recent techniques are based on least-squares estimation methods and obviate the need for input whiteness or Gaussianness, although they may still impose certain restrictions (e.g., sufficiently broadband inputs are required and the estima ted models must be complete). The Laguerre expansion technique (Marmarelis, 1993) belongs to this latter category and overcomes the limitations imposed by the crosscorrelation technique in the case of non-Gaussian or non-white inputs, allowing accurate kernel estimation under previously prohibitive conditions (e.g., diagonal kernel values can now be estimated with binary inputs). This paper demonstrates the expanded capabilities of kernel estimation using Laguerre expansions when the input is non-Gaussian and/or non-white, enlarging the scope of Volterra-Wiener analysis in actual applications.

Keywords

Kernel Estimate Volterra Kernel Cross Correlation Technique Binary Input Wiener Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Barker, H.A. and Pradisthayon T. (1970) “High-order autocorrelation functions of pseudorandom signals based on M-sequences” Proc. IEE, 117:1857–1863.Google Scholar
  2. 2.
    Goussard, Y. (1987) Wiener Kernel Estimation: A comparison of cross correlation and stochastic approximation methods. In-Advanced Methods of Physiological System Modeling, Vol. I, V.Z. Marmarelis (ed.), Biomedical Simulations Resource, University of Southern California, Los Angeles, California, pp. 289–302.Google Scholar
  3. 3.
    Gyftopoulos, E.P., and Hooper R.J. (1964) Signals for transfer function measurement in nonlinear systems, Noise Analysis in Nuclear Systems, USAEC Symposium Series 4, TID-7679.Google Scholar
  4. 4.
    Klein, S. and Yasui, S. (1979) Nonlinear systems analysis with non-Gaussian white stimuli: General basis functionals and kernels. IEEE Trans. Info. Theo., 25:495–500.zbMATHCrossRefGoogle Scholar
  5. 5.
    Korenberg, M.J. (1987) Functional expansions, parallel cascades and nonlinear difference equations. In-Advanced Methods of Physiological System Modeling, Vol. I, V.Z. Marmarelis (ed.), Biomedical Simulations Resource, University of Southern California, Los Angeles, California, pp. 221–240.Google Scholar
  6. 6.
    Lee, Y.W. and Schetzen, M. (1965) Measurement of the Wiener kernels of a nonlinear system by crosscorrelation. Int. J. Contr., 2:237–254.CrossRefGoogle Scholar
  7. 7.
    Marmarelis, P.Z. and Marmarelis, V.Z. (1978) Analysis of Physiological Systems: The White-noise Approach, Plenum, New York, New York.CrossRefGoogle Scholar
  8. 8.
    Marmarelis, V.Z. (ed.) (1987) Advanced Methods of Physiological System Modeling, Vol. I, Biomedical Simulations Resource, University of Southern California, Los Angeles, California.Google Scholar
  9. 9.
    Marmarelis, V.Z. (ed.) (1989) Advanced Methods of Physiological System Modeling, Vol. II, Plenum, New York, New York.Google Scholar
  10. 10.
    Marmarelis, V.Z. (1975) Identification of nonlinear systems through multi-level random signals. Proc. 1st Symp. on Testing and Identification of Nonlinear Systems, Pasadena, California, pp. 106–124.Google Scholar
  11. 11.
    Marmarelis, V.Z. (1977) A family of quasi-white random signals and its optimal use in biological system identification. Part I: Theory. Biol. Cyb., 27:49–56.zbMATHCrossRefGoogle Scholar
  12. 12.
    Marmarelis, V.Z. (1978) Random vs. pseudorandom test signals in nonlinear system identification. Proc. IEE, 125:425–428.Google Scholar
  13. 13.
    Marmarelis, V.Z. (1979) Error analysis and optimal estimation procedures in identification of nonlinear Volterra systems. Automatica, 15:161–174.zbMATHCrossRefGoogle Scholar
  14. 14.
    Marmarelis, V.Z. (1987) Nonlinear and nonstationary modeling of physiological systems. In-Advanced Models of Physiological System Modeling, Vol. I, V.Z. Marmarelis (ed.), Biomedical Simulations Resource, University of Southern California, Los Angeles, California, pp.1–24.Google Scholar
  15. 15.
    Marmarelis, V.Z. (1993) Identification of nonlinear biological systems using Laguerre expansions of kernels. Ann. Biomed. Eng., 21:573–589.CrossRefGoogle Scholar
  16. 16.
    Marmarelis, V.Z., and McCann, G.D. (1977) A family of quasi-white random signals and its optimal use in biological system identification. Part II: Application to the photoreceptor of Calliphora Erythrocephala. Biol. Cyb., 27:57–62.CrossRefGoogle Scholar
  17. 17.
    Marmarelis, V.Z., Chon, K.H., Chen, Y.M., Marsh, D.J. and Holstein-Rathlou, N.H. (1993) Nonlinear analysis of renal autoregulation under broadband forcing conditions. Ann. Biomed. Eng., 21:591–603.CrossRefGoogle Scholar
  18. 18.
    Ream, N. (1970) Nonlinear identification using inverse-repeat m-sequences. Proc. IEE, 117:213–218.MathSciNetGoogle Scholar
  19. 19.
    Sutter, E.E. (1987) A practical nonstochastic approach to nonlinear time-domain analysis. In: Advanced Methods of Physiological System Modeling, Vol. I, V.Z. Marmarelis (ed.), Biomedical Simulations Resourse, University of Southern Califonia, Los Angeles, California, pp. 303–315.Google Scholar
  20. 20.
    Victor, J.D. (1979) Nonlinear systems analysis: Comparison of white noise and sum of sinusoids in a biological system. Proc. Nat. Acad. Sei., 76:996–998.CrossRefGoogle Scholar
  21. 21.
    Wiener, N. (1958) Nonlinear Problems in Random Theory, Wiley, New York, New York.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Vasilis Z. Marmarelis
    • 1
  1. 1.Departments of Biomedical and Electrical EngineeringUniversity of Southern CaliforniaUSA

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