OMNE: A new robust membership-set estimator for the parameters of nonlinear models

  • Hélène Lahanier
  • Eric Walter
  • Roberto Gomeni


A new method for estimating parameters and their uncertainty is presented. Data are assumed to be corrupted by a noise whose statistical properties are unknown but for which bounds are available at each sampling time. The method estimates the set of all parameter vectors consistent with this hypothesis. Its results are compared with those of the weighted least squares, extended least squares, and biweight robust regression approaches on two data sets, one of which includes 33% outliers. On the basis of these preliminary results, the new method appears to have attractive properties of reliability and robustness.

Key words

Outliers robust estimation uncertainty intervals membership set extended least squares biweight regression weighted least squares 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. C. Peck, S. L. Beal, L. B. Sheiner, and A. I. Nichols. Extended least squares nonlinear regression: A possible solution to the “choice of weight” problem in analysis of individual pharmacokinetic data.J. Pharmacokin. Biopharm. 12:545–558 (1984).CrossRefGoogle Scholar
  2. 2.
    G. E. Box and W. J. Hill. Correcting inhomogeneity of variance with power transformation weighting.Technometrics 16:385–389 (1974).CrossRefGoogle Scholar
  3. 3.
    L. B. Sheiner. Analysis of pharmacokinetic data using parametric models. 1: Regression models.J. Pharmacokin. Biopharm. 12:93–117 (1984).CrossRefGoogle Scholar
  4. 4.
    R. L. Launer and G. N. Wilkinson (eds.).Robustness in Statistics, Academic Press, New York, 1979.Google Scholar
  5. 5.
    A. Cornish-Bowden. Robust estimation in enzyme kinetics. In L. Endrenyi (ed.),Kinetic Data Analysis, Plenum Press, New York, 1981, pp. 105–135.CrossRefGoogle Scholar
  6. 6.
    E. Fogel. System identification via membership set constraints with energy constrained noise.IEEE Trans. Automatic Control AC-24:752–758 (1979).CrossRefGoogle Scholar
  7. 7.
    M. Milanese and G. Belforte. Estimation theory and uncertainty intervals evaulation in presence of unknown but bounded errors: Linear families of models and estimators.IEEE Trans. Automatic Control AC-27:408–414 (1982).CrossRefGoogle Scholar
  8. 8.
    F. C. Schweppe. Recursive state estimation: Unknown but bounded errors and system inputs.IEEE Trans. Automatic Control AC-13:22–28 (1968).CrossRefGoogle Scholar
  9. 9.
    D. P. Bertsekas and I. B. Rhodes. Recursive state estimation for a set membership description of uncertainty.IEEE Trans. Automatic Control AC-16:117–128 (1971).CrossRefGoogle Scholar
  10. 10.
    G. A. Bekey and S. F. Masri. Random search techniques for optimization of nonlinear systems with many parameters.Math. Comput. Simulation 25:(3):210–213 (1983).CrossRefGoogle Scholar
  11. 11.
    L. Pronzato, E. Walter, A. Venot, and J. F. Lebruchec. A general purpose global optimizer: Implementations and applications.Math. Comput. Simulation 26:412–422 (1984).CrossRefGoogle Scholar
  12. 12.
    J. Richalet, A. Rault, and R. Poliquen.Identification des processus par la méthods du modéle, Gordon & Breach, Paris, 1971, pp. 177–197.Google Scholar
  13. 13.
    B. T. Poljak and J. Z. Tsypkin. Robust identification.Automatica 16:53–63 (1980).CrossRefGoogle Scholar
  14. 14.
    A. Venot, L. Pronzato, E. Walter, and J. F. Lebruchec. A distribution-free criterion for robust identification, with applications in system modelling and image processing.Automatica 22:105–109 (1986).CrossRefGoogle Scholar
  15. 15.
    R. V. Hogg. An introduction to robust estimation. In R. L. Launer and G. N. Wilkinson (eds.),Robustness in Statistics, Academic Press, New York, 1979, pp. 1–17.Google Scholar
  16. 16.
    R. Gomeni, H. Lahanier, and E. Walter. Study of the pharmacokinetics of betaxolol using membership set estimation, InProceedings of the Third IMEKO Congress on Measurement in Clinical Medicine, Edinburgh, 9–11 September 1986, pp. 229–240.Google Scholar
  17. 17.
    E. Walter, H. Lahanier, and J. Happel. Estimation of non uniquely identifiable parameters via exhaustive modeling and membership set theory. InProceedings of the First IMACS/IFAC Congress on Modelling and Simulation for Control of Lumped and Distributed Parameter Systems, Lille, 3–6 June 1986, pp. 313–317.Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • Hélène Lahanier
    • 1
  • Eric Walter
    • 1
  • Roberto Gomeni
    • 2
  1. 1.Laboratoire des Signaux et SystèmesCNRS Ecole Supérieure d'ElectricitéGif-Sur-YvetteFrance
  2. 2.Centre d'Etudes et de Recherches en Statistiques et Informatique MédicalesSIMEDCréteilFrance

Personalised recommendations