Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-15T09:22:33.430Z Has data issue: false hasContentIssue false
  • English
  • Français

Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE

Published online by Cambridge University Press:  28 July 2014

E. Grenier ,
V. Louvet  and
P. Vigneaux
E. Grenier
Affiliation:
U.M.P.A., Ecole Normale Supérieure de Lyon, CNRS UMR 5669 & INRIA, Project-team NUMED. 46 Allée d’Italie, 69364 Lyon Cedex 07, France. [email protected]
V. Louvet
Affiliation:
Institut Camille Jordan, CNRS UMR 5208 & Université Lyon 1 & INRIA, Project-team NUMED. 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
P. Vigneaux
Affiliation:
U.M.P.A., Ecole Normale Supérieure de Lyon, CNRS UMR 5669 & INRIA, Project-team NUMED. 46 Allée d’Italie, 69364 Lyon Cedex 07, France. [email protected]
Get access

Abstract

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases impossible to do within a reasonable time. We propose here a very simple methodology which may accelerate population parametrization of complex models, including partial differential equations models. We illustrate our method on the classical KPP equation.

Keywords

Parameter estimationSAEM algorithmpartial differential equationsKPP equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1303 - 1329
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H.T. Banks and K. Kunisch, Estimation techniques for distributed parameter systems, vol. 1. Systems & Control: Foundations & Appl. Birkhäuser Boston Inc., Boston, MA (1989).
Brauns, G.T., Bishop, R.J., Steer, M.B., Paulos, J.J. and Ardalan, S.H., Table-based modeling of delta-sigma modulators using Zsim. IEEE Trans. Computer-Aided Design Integr. Circuits Syst. 9 (1990) 142–150. Google Scholar
Delyon, B., Lavielle, M. and Moulines, E., Convergence of a stochastic approximation version of the EM algorithm. Ann. Statis. 27 94–128 (1999). Google Scholar
Donnet, S., Foulley, J.-L. and Samson, A., Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations. Biometrics 66 (2010) 733–741. Google ScholarPubMed
Donnet, S. and Samson, A., Parametric inference for mixed models defined by stochastic differential equations. ESAIM: PS 12 (2008) 196–218. Google Scholar
R.A. Fisher, The Genetical Theory of Natural Selection. Oxford University Press (1930).
Giles, M.B. and Pierce, N.A., An introduction to the adjoint approach to design. Flow, Turbulence and Combustion 65 (2000) 393–415. Google Scholar
Goughran, W.M., Grosse, E. and Rose, D.J., CAzM: A circuit analyzer with macromodeling. IEEE Trans. Electron. Devices 30 (1983) 1207–1213. Google Scholar
V. Isakov, Inverse problems for partial differential equations, vol. 127. Appl. Math. Sci., 2nd edition. Springer, New York (2006).
J. Kaipio and E. Somersalo, Statistical and computational inverse problems, vol. 160. Appl. Math. Sci. Springer-Verlag, New York (2005).
Kolmogoroff, A., Petrovsky, I. and Piscounoff, N., Étude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un problème biologique. Bulletin de l’université d’État à Moscou, Section A I (1937) 1–26. Google Scholar
Kuhn, E. and Lavielle, M., Maximum likelihood estimation in nonlinear mixed effects models. Comput. Statis. Data Anal. 49 (2005) 1020–1038. Google Scholar
M. Lavielle, Private Communication (2012).
M. Lavielle and K. Bleakley, Population Approach & Mixed Effects Models – Models, Tasks, Tools & Methods. Avalaible at http://popix.lixoft.net/ INRIA (2013).
Lavielle, M. and Mentré, F., Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the monolix software. J. Pharmacokinetics and Pharmacodynamics 34 (2007) 229–249. Google Scholar
J.-L. Lions, Optimal control of systems governed by partial differential equations. Translated from the French by S.K. Mitter. Springer-Verlag, New York (1971).
Nie, L., Strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models. Metrika 63 (2006) 123–143. Google Scholar
Nie, L. and Yang, M., Strong consistency of mle in nonlinear mixed-effects models with large cluster size. Sankhya: Indian J. Statis. 67 (2005) 736–763. Google Scholar
Snoeck, E., Chanu, P., Lavielle, M., Jacqmin, P., Jonsson, E.N., Jorga, K., Goggin, T., Grippo, J., Jumbe, N.L. and Frey, N., A Comprehensive Hepatitis C Viral Kinetic Model Explaining Cure. Clinical Pharmacology & Therapeutics 87 (2010) 706–713. Google Scholar
A. Tarantola, Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005).
M. Team, The Monolix software, Version 4.1.2. Analysis of mixed effects models. Available at http://www.lixoft.com/ LIXOFT and INRIA (2012).
Yu, G. and Li, P., Efficient look-up-table-based modeling for robust design of sigma-delta ADCs. IEEE Trans. Circuits Syst. – I 54 (2007) 1513–1528. Google Scholar