Abstract
Many techniques have been developed for the estimation of the Volterra/Wiener kernels of nonlinear systems, and have found extensive application in the study of various physiological systems. To date, however, we are not aware of methods for estimating the reliability of these kernels from single data records. In this study, we develop a formal analysis of variance for least-squares based nonlinear system identification algorithms. Expressions are developed for the variance of the estimated kernel coefficients and are used to place confidence bounds around both kernel estimates and output predictions. Specific bounds are developed for two such identification algorithms: Korenberg's fast orthogonal algorithm and the Laguerre expansion technique. Simulations, employing a model representative of the peripheral auditory system, are used to validate the theoretical derivations, and to explore their sensitivity to assumptions regarding the system and data. The simulations show excellent agreement between the variances of kernel coefficients and output predictions as estimated from the results of a single trial compared to the same quantities computed from an ensemble of 1000 Monte Carlo runs. These techniques were validated with white and nonwhite Gaussian inputs and with white Gaussian and nonwhite non-Gaussian measurement noise on the output, provided that the output noise source was independent of the test input.
Similar content being viewed by others
REFERENCES
Amorocho, J., and A. Brandstetter. Determination of nonlinear functional response functions in rainfall runoff processes. Water Resour. Res.7:1087-1101, 1971.
Beck, J. V., and K. J. Arnold. Parameter Estimation in Engineering and Science. New York: Wiley, 1977, pp. 213- 333.
Billings, S. A., and W. S. F. Voon. Correlation based model validity tests for non-linear models. Int. J. Control44:235- 244, 1986.
Boyd, S., and L. O. Chua. Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst.CAS-32:1150-1161, 1985.
Chon, K. H., Y. M. Chen, V. Z. Marmarelis, D. J. Marsh, and N. H. Holstein-Rathlou. Detection of interactions between myogenic and TGF mechanisms using nonlinear analysis. Am. J. Physiol.267:F160-F173, 1994.
Chon, K. H., N.-H. Holstein-Rathlou, D. J. Marsh, and V. Z. Marmarelis. Parametric and nonparametric nonlinear modeling of renal autoregulation dynamics. In: Advanced Methods of Physiological System Modeling, Vol. 3, edited by V. Z. Marmarelis. New York: Plenum, 1994, pp. 195-210.
Efron, B. The Jackknife, the Bootstrap and Other Resampling Plans. Philadelphia: Society for Industrial and Applied Mathematics, 1982, pp. 29-35.
Emerson, R. C., M. J. Korenberg, and M. C. Citron. Identi-fication of complex-cell intensive nonlinearities in a cascade model of cat visual cortex. Biol. Cybern.66:291-300, 1992.
French, A. S., and M. J. Korenberg. Disection of a nonlinear cascade model for sensory encoding. Ann. Biomed. Eng.19:473-484, 1991.
French, A. S., A. E. C. Pece, and M. J. Korenberg. Nonlinear models of transduction and adaptation in Locust photoreceptors. In: Advanced Methods of Physiological System Modeling, Vol. 2, edited by V. Z. Marmarelis. New York: Plenum, 1989, pp. 81-95.
Hunter, I. W., and M. J. Korenberg. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern.55:135-144, 1986.
Korenberg, M. J. Functional expansions, parallel cascades, and nonlinear difference equations. In: Advanced Methods of Physiological System Modeling, Vol. 1, edited by V. Z. Marmarelis. New York: Plenum, 1987, pp. 221-240.
Korenberg, M. J. Identifying nonlinear difference equation and functional expansion representations: The fast orthogonal algorithm. Ann. Biomed. Eng.16:123-142, 1988.
Korenberg, M. J. Parallel cascade identification and kernel estimation for nonlinear systems. Ann. Biomed. Eng.19:429- 455, 1991.
Korenberg, M. J., and I. W. Hunter. The identification of nonlinear biological systems: LNL cascade models. Biol. Cybern.55:125-134, 1986.
Korenberg, M. J., and I. W. Hunter. The identification of nonlinear biological systems: Wiener kernel approaches. Ann. Biomed. Eng.18:629-654, 1990.
Korenberg, M. J., and I. W. Hunter. The identification of nonlinear biological systems: Volterra kernel approaches. Ann. Biomed. Eng.24:250-268, 1996.
Korenberg, M. J., H. M. Sakai, and K.-I. Naka. Dissection of the neuron network in the catfish inner retina. III interpretation of spike kernels. J. Neurophysiol.61:1110-1120, 1989.
Lee Y. W., and M. Schetzen. Measurement of the Wiener kernels of a non-linear system by cross-correlation. Int. J. Control2:237-254, 1965.
Ljung, L. System Identification: Theory for the User. Englewood Cliffs: Prentice-Hall, 1987, pp. 434-456.
Marmarelis, V. Z. Identification of nonlinear biological systems using Laguerre expansions of kernels. Ann. Biomed. Eng.21:573-589, 1993.
Marmarelis, V. Z., K. H. Chon, Y. M. Chen, D. J. Marsh, and N. H. Holstein-Rathlou. Nonlinear analysis of renal autoregulation under broadband forcing conditions. Ann. Biomed. Eng.21:591-603, 1993.
Marmarelis, P. Z., and V. Z. Marmarelis. Analysis of Physiological Systems. New York: Plenum, 1978, pp. 1-487.
Pfeiffer, R. R. A model for two-tone inhibition of single cochlear nerve fibres. J. Acoust. Soc. Am.48:1373-1378, 1970.
Sakuranaga, M., S. Sato, E. Hida, and K-I. Naka. Nonlinear analysis: mathematical theory and biological applications. CRC Crit. Rev. Biomed. Eng.14:127-184, 1986.
Swerup, C. On the choice of noise for the analysis of the peripheral auditory system. Biol. Cybern.29:97-104, 1978.
Watanabe, A., and L. Stark. Kernel method for nonlinear analysis: identification of a biological control system. Math. Biosci.27:99-108, 1975.
Zhang, Q., B. Suki, D. T. Westwick, and K. R. Lutchen. Factors affecting Volterra kernel estimation: Emphasis on lung tissue viscoelasticity. Ann. Biomed. Eng. 26:103-116.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Westwick, D.T., Suki, B. & Lutchen, K.R. Sensitivity Analysis of Kernel Estimates: Implications in Nonlinear Physiological System Identification. Annals of Biomedical Engineering 26, 488–501 (1998). https://doi.org/10.1114/1.40
Issue Date:
DOI: https://doi.org/10.1114/1.40