Advertisement

Journal of Mathematical Biology

, Volume 28, Issue 5, pp 501–527 | Cite as

Statistical methods for model comparison in parameter estimation problems for distributed systems

  • H. T. Banks
  • B. G. Fitzpatrick
Article

Abstract

In this note we outline some recent results on the development of a statistical testing methodology for inverse problems involving partial differential equation models. Applications to several problems from biology are presented. The statistical tests, which are in the spirit of analysis of variance (ANOVA), are based on asymptotic distributional results for estimators and residuals in a least squares approach.

Key words

Hypothesis testing Parameter estimation Partial differential equations Approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B1]
    Banks H. T.: On a variational approach to some parameter estimation problems. In: Kappel, F., Kunisch, K., Schappacher, W. (eds.) Distributed parameter systems. Proceedings, Vorau, Austria 1985. (Lect. Notes Control Info. Sci., vol. 75, pp. 9–23) Berlin Heidelberg, New York: Springer 1985.Google Scholar
  2. [B2]
    Banks, H. T.: Computational techniques for inverse problems in size structured stochastic population models, LCDS-CCS Report 87–41, Division of Applied Mathematics, Brown University, Providence, RI (1987); and in: Bermudez, A. (ed.), Control of Partial Differential Equations, Proceedings, Santiago de Compostela, Spain 1987. (Lect. Notes Control Inf. Sci., vol. 114, pp. 3–10) Berlin Heidelberg New York: Springer 1988.Google Scholar
  3. [BBKW]
    Banks, H. T., Botsford, L. W., Kappel F., Wang, C.: Modeling and estimation in size structured population models. LCDS-CCS Report 87–13, Division of Applied Mathematics, Brown University, Providence, RI (1987), and in: Proceedings 2nd Course on Mathematical Ecology, Trieste, 1986, pp. 521–541. Singapore: World Press 1988.Google Scholar
  4. [BCK]
    Banks, H. T., Crowley, J. M., Kunisch, K.: Cubic spline approximation techniques for parameter estimation in distributed systems. IEEE Trans. Autom. Control AC-28, 773–786 (1983).Google Scholar
  5. [BCR]
    Banks, H. T., Crowley, J. M., Rosen, I. G.: Methods for identification of material parameters in distributed models for flexible structures. Mat. Apl. Comput. 5, 139–168 (1986).Google Scholar
  6. [BFW]
    Banks, H. T., Fabiano, R., Wang, Y.: Estimation of Boltzmann damping coefficients in beam models, LCDS-CCS Report 88–13, Division of Applied Mathematics, Brown University, Providence, RI (1988) and in: Balakrishnan, A. V., Zolesio, J. P. (eds.), COMCON Conference on Stabilization of Flexible Structures, pp. 13–35. New York: Optimization Software 1988.Google Scholar
  7. [BI]
    Banks, H. T., Ito, K.: A unified framework for approximation and inverse problems for distributed parameter systems. Control Theory Adv. Technol. 4, 73–90 (1988).Google Scholar
  8. [B12]
    Banks, H. T., Iles, D. W.: On compactness of admissible parameter sets: Convergence and stability in inverse problems for distributed parameter systems. ICASE Report 86–83, NASA Langley Research Center, Hampton, VA, 1986.Google Scholar
  9. [BK1]
    Banks, H. T., P. Kareiva, Parameter estimation techniques for transport equations with applications to population dispersal and tissue bulk flow models. J. Math. Biol. 17, 253–272 (1983).Google Scholar
  10. [BKL]
    Banks, H. T., Karieva, P., and Lamm, P. D.: Modeling insect dispersal and estimating parameters when mark-release techniques may cause initial disturbances. J. Math. Biol. 22, 259–277 (1985).Google Scholar
  11. [BKM]
    Banks, H. T., Kareiva, P., Murphy, K.: Parameter estimation techniques for interaction and redistribution models: a predator-prey example. Oecologia 74, 356–362 (1987).Google Scholar
  12. [BKZ]
    Banks, H. T., Kareiva, P., Zia, L.: Analyzing field studies of insect dispersal using two-dimensional transport equations. Environ. Entomol. 17, 815–820 (1988).Google Scholar
  13. [BK2]
    Banks, H. T., Kunisch, K.: Estimation techniques for distributed parameter systems. Boston: Birkhäuser 1989.Google Scholar
  14. [BL]
    Banks, H. T., Lamm, P. D.: Estimation of variable coefficients in parabolic distributed systems. IEEE Trans. Autom. Control AC-30, 386–398 (1985).Google Scholar
  15. [BRR]
    Banks, H. T., Reich, S., Rosen, I.G.: An approximation theory for the identification of nonlinear distributed parameter systems, LCDS-CCS Report 88–8, Division of Applied Mathematics, Brown University, Providence, RI (1988) and SIAM J. Cont. Optimization, to appear.Google Scholar
  16. [BR]
    Banks, H. T., Rosen, I. G.: Numerical schemes for the estimation of functional parameters in distributed models for mixing mechanisms in lake and sea sediment cores. Inverse Problems 3, 1–23 (1987).Google Scholar
  17. [BWIC]
    Banks, H. T., Wang, Y., Inman, D. J., Cudney, H.: Parameter identification techniques for the estimation of damping in flexible structure experiments, pp. 1392–1395. Proc. 20th IEEE Conf. Dec. and Control, Los Angeles (1987).Google Scholar
  18. [B3]
    Billingsley, P.: Convergence of probability measures. New York: Wiley 1968.Google Scholar
  19. [B4]
    Billingsley, P.: Probability and measure. New York: Wiley 1979.Google Scholar
  20. [DBW]
    Dexter, F., Banks, H. T., Webb III, T.: Modeling holocene changes in the location and abundance of beech populations in eastern North America. Rev. Palaeobot. Palynol. 50, 273–292 (1987).Google Scholar
  21. [F]
    Fitzpatrick, B. G.: Statistical methods in parameter identification and model selection. Ph.D. Thesis, Division of Applied Mathematics, Brown University, Providence, RI, 1988.Google Scholar
  22. [GI]
    Gallant, A. R.: Nonlinear statistical models. New York: Wiley 1987.Google Scholar
  23. [G2]
    Graybill, F.: Theory and application of the linear model. North Scituate, Mass.: Duxbury 1976.Google Scholar
  24. [KN]
    Kitamura, S., Nakagiri, S.: Identifiability of spatially varying and constant parameters in distributed systems of parabolic type. SIAM J. Control Optimization 15, 785–802 (1977).Google Scholar
  25. [L]
    Lehmann, E. L.: Testing statistical hypotheses. New York: Wiley 1959.Google Scholar
  26. [MD]
    Metz, J. A. J., Diekmann, O.: The dynamics of physiologically structured populations (Lect. Notes Biomath., vol. 68) Berlin Heidelberg New York: Springer 1986.Google Scholar
  27. [O]
    Okubo, A.: Diffusion and ecological problems: Mathematical models. Berlin Heidelberg New York: Springer 1980.Google Scholar
  28. [R]
    Royden, H. L.: Real analysis. New York: MacMillan 1968.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • H. T. Banks
    • 1
  • B. G. Fitzpatrick
    • 1
  1. 1.Center for Control Sciences, Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations