Population parametrization of costly black box models using iterations between SAEM algorithm and Kriging. (English) Zbl 1396.65015
Summary: In this article we focus on parametrization of black box models from repeated measurements among several individuals (population parametrization). We introduce a variant of the SAEM algorithm, called KSAEM algorithm, which couples the standard SAEM algorithm with the dynamic construction of an approximate metamodel. The costly evaluation of the genuine black box is replaced by a kriging step, using a basis of precomputed values, basis which is enlarged during SAEM algorithm to improve the accuracy of the metamodel in regions of interest.
MSC:
| 65C60 | Computational problems in statistics (MSC2010) |
| 65C40 | Numerical analysis or methods applied to Markov chains |
| 62M05 | Markov processes: estimation; hidden Markov models |
| 60G15 | Gaussian processes |
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
Keywords:
parameters estimation; SAEM algorithm; Kriging; non-linear mixed effect models; partial differential equations; KPP equationReferences:
| [1] | Auder, B; Crecy, A; Iooss, B; Marques, M, Screening and metamodeling of computer experiments with functional outputs, Reliab Eng Syst Saf, 107, 122-131, (2012) · doi:10.1016/j.ress.2011.10.017 |
| [2] | Barbillon P, Barthelemy C, Samson A (2015) Parametric estimation of complex mixed models based on meta-model approach (submitted) · Zbl 1384.62062 |
| [3] | Delyon, B; Lavielle, M; Moulines, E, Convergence of a stochastic approximation version of the EM algorithm, Ann Stat, 27, 94-128, (1999) · Zbl 0932.62094 · doi:10.1214/aos/1018031103 |
| [4] | Dempster, AP; Laird, NM; Rubin, DB, Maximum likelihood from incomplete data via the EM algorithm, J R Stat Soc Ser B Methodol, 39, 1-38, (1977) · Zbl 0364.62022 |
| [5] | Dupuy, D; Helbert, C; Franco, J, Dicedesign and diceeval: two R packages for design and analysis of computer experiments, J Stat Softw, 65, 1-38, (2015) · doi:10.18637/jss.v065.i11 |
| [6] | Fang K, Li R, Sudjianto A (2005) Design and modeling for computer experiments (Computer science and data analysis). Chapman & Hall/CRC, London · Zbl 1093.62117 |
| [7] | Grenier, E; Louvet, V; Vigneaux, P, Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE, ESAIM: M2AN, 48, 1303-1329, (2014) · Zbl 1301.35177 · doi:10.1051/m2an/2013140 |
| [8] | Haasdonk, B; Lohmann, B, Special issue on model order reduction of parameterized problems, Math Comput Model Dyn Syst, 17, 295-296, (2011) · Zbl 1298.00140 · doi:10.1080/13873954.2011.547661 |
| [9] | Jin, R; Chen, W; Sudjianto, A, An efficient algorithm for constructing optimal design of computer experiments, J Stat Plan Inference, 134, 268-287, (2005) · Zbl 1066.62075 · doi:10.1016/j.jspi.2004.02.014 |
| [10] | Kuhn, E; Lavielle, M, Maximum likelihood estimation in nonlinear mixed effects models, Comput Stat Data Anal, 49, 1020-1038, (2005) · Zbl 1429.62279 · doi:10.1016/j.csda.2004.07.002 |
| [11] | Marrel, A; Iooss, B; Julien, M; Laurent, B; Volkova, E, Global sensitivity analysis for models with spatially dependent outputs, Environmetrics, 22, 383-397, (2011) · doi:10.1002/env.1071 |
| [12] | Monolix Team (2012) The Monolix software, version 4.1.2. Analysis of mixed effects models. See “The theophylline example” section in the PDF of the Users Guide. http://www.lixoft.eu/wp-content/resources/docs/UsersGuide.pdf, LIXOFT and INRIA. http://www.lixoft.com |
| [13] | Moutoussamy, V; Nanty, S; Pauwels, B, Emulators for stochastic simulation codes, ESAIM Proc Surv, 48, 116-155, (2015) · Zbl 1338.62179 · doi:10.1051/proc/201448005 |
| [14] | Patera A, Rozza G (2006) Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. MIT Monographs (also available online) |
| [15] | Picheny, V; Ginsbourger, D, A non-stationary space-time Gaussian process model for partially converged simulations, J Uncertain Quantif, 1, 57-78, (2013) · Zbl 1291.60075 · doi:10.1137/120882834 |
| [16] | Pronzato, L; Muller, W, Design of computer experiments: space filling and beyond, Stat Comput, 22, 681-701, (2012) · Zbl 1252.62080 · doi:10.1007/s11222-011-9242-3 |
| [17] | Sacks, J; Schiller, SB; Mitchell, TJ; Wynn, HP, Design and analysis of computer experiments, Stat Sci, 4, 409-435, (1989) · Zbl 0955.62619 · doi:10.1214/ss/1177012413 |
| [18] | Santner TJ, Williams B, Notz W (2003) The design and analysis of computer experiments. Springer, Berlin · Zbl 1041.62068 · doi:10.1007/978-1-4757-3799-8 |
| [19] | Schilders WHA, van der Vorst HA (eds) (2008) Model order reduction: theory, research aspects and applications. Springer, Berlin · Zbl 1154.93322 |
| [20] | Wei, G; Tanner, M, A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms, J Am Stat Assoc, 85, 699-704, (1990) · doi:10.1080/01621459.1990.10474930 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
