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

E. Grenier; V. Louvet; P. Vigneaux

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 5, page 1303-1329
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Grenier, E., Louvet, V., and Vigneaux, P.. "Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1303-1329. <http://eudml.org/doc/273171>.

@article{Grenier2014,
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.},
author = {Grenier, E., Louvet, V., Vigneaux, P.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {parameter estimation; SAEM algorithm; partial differential equations; KPP equation; SAEM (stochastic approximation expectation-maximization) algorithm; (Kolmogorov-Petrovskii-Piscunov) KPP equation},
language = {eng},
number = {5},
pages = {1303-1329},
publisher = {EDP-Sciences},
title = {Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE},
url = {http://eudml.org/doc/273171},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Grenier, E.
AU - Louvet, V.
AU - Vigneaux, P.
TI - Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1303
EP - 1329
AB - 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.
LA - eng
KW - parameter estimation; SAEM algorithm; partial differential equations; KPP equation; SAEM (stochastic approximation expectation-maximization) algorithm; (Kolmogorov-Petrovskii-Piscunov) KPP equation
UR - http://eudml.org/doc/273171
ER -

References

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  1. [1] H.T. Banks and K. Kunisch, Estimation techniques for distributed parameter systems, vol. 1. Systems & Control: Foundations & Appl. Birkhäuser Boston Inc., Boston, MA (1989). Zbl0695.93020MR1045629
  2. [2] G.T. Brauns, R.J. Bishop, M.B. Steer, J.J. Paulos and S.H. Ardalan, Table-based modeling of delta-sigma modulators using Zsim. IEEE Trans. Computer-Aided Design Integr. Circuits Syst.9 (1990) 142–150. 
  3. [3] B. Delyon, M. Lavielle and E. Moulines, Convergence of a stochastic approximation version of the EM algorithm. Ann. Statis.2794–128 (1999). Zbl0932.62094MR1701103
  4. [4] S. Donnet, J.-L. Foulley and A. Samson, Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations. Biometrics66 (2010) 733–741. Zbl1203.62187MR2758209
  5. [5] S. Donnet and A. Samson, Parametric inference for mixed models defined by stochastic differential equations. ESAIM: PS 12 (2008) 196–218. Zbl1182.62164MR2374638
  6. [6] R.A. Fisher, The Genetical Theory of Natural Selection. Oxford University Press (1930). Zbl56.1106.13MR1785121JFM56.1106.13
  7. [7] M.B. Giles and N.A. Pierce, An introduction to the adjoint approach to design. Flow, Turbulence and Combustion 65 (2000) 393–415. Zbl0996.76023
  8. [8] W.M. Goughran, E. Grosse and D.J. Rose, CAzM: A circuit analyzer with macromodeling. IEEE Trans. Electron. Devices30 (1983) 1207–1213. 
  9. [9] V. Isakov, Inverse problems for partial differential equations, vol. 127. Appl. Math. Sci., 2nd edition. Springer, New York (2006). Zbl1092.35001MR2193218
  10. [10] J. Kaipio and E. Somersalo, Statistical and computational inverse problems, vol. 160. Appl. Math. Sci. Springer-Verlag, New York (2005). Zbl1068.65022MR2102218
  11. [11] A. Kolmogoroff, I. Petrovsky and N. Piscounoff, 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. Zbl0018.32106
  12. [12] E. Kuhn and M. Lavielle, Maximum likelihood estimation in nonlinear mixed effects models. Comput. Statis. Data Anal.49 (2005) 1020–1038. Zbl05374202MR2143055
  13. [13] M. Lavielle, Private Communication (2012). 
  14. [14] M. Lavielle and K. Bleakley, Population Approach & Mixed Effects Models – Models, Tasks, Tools & Methods. Avalaible at http://popix.lixoft.net/ INRIA (2013). 
  15. [15] M. Lavielle and F. Mentré, Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the monolix software. J. Pharmacokinetics and Pharmacodynamics34 (2007) 229–249. 
  16. [16] 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). Zbl0203.09001
  17. [17] L. Nie, Strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models. Metrika63 (2006) 123–143. Zbl1095.62028
  18. [18] L. Nie and M. Yang, Strong consistency of mle in nonlinear mixed-effects models with large cluster size. Sankhya: Indian J. Statis. 67 (2005) 736–763. Zbl1193.62028
  19. [19] E. Snoeck, P. Chanu, M. Lavielle, P. Jacqmin, E.N. Jonsson, K. Jorga, T. Goggin, J. Grippo, N.L. Jumbe and N. Frey, A Comprehensive Hepatitis C Viral Kinetic Model Explaining Cure. Clinical Pharmacology & Therapeutics 87 (2010) 706–713. 
  20. [20] A. Tarantola, Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2005). Zbl1074.65013MR2130010
  21. [21] M. Team, The Monolix software, Version 4.1.2. Analysis of mixed effects models. Available at http://www.lixoft.com/ LIXOFT and INRIA (2012). 
  22. [22] G. Yu and P. Li, Efficient look-up-table-based modeling for robust design of sigma-delta ADCs. IEEE Trans. Circuits Syst. – I54 (2007) 1513–1528. 

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