Adaptive control for sequential design

Roland Gautier; Luc Pronzato

Discussiones Mathematicae Probability and Statistics (2000)

  • Volume: 20, Issue: 1, page 97-114
  • ISSN: 1509-9423

Abstract

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The optimal experiment for estimating the parameters of a nonlinear regression model usually depends on the value of these parameters, hence the problem of designing experiments that are robust with respect to parameter uncertainty. Sequential designpermits to adapt the experiment to the value of the parameters, and can thus be considered as a robust design procedure. By designing theexperiments sequentially, one introduces a feedback of information, and thus dynamics, into the design procedure. Several sequential schemes, corresponding to different control policies, are considered. The optimal one corresponds to closed-loop control, and is solution of a stochastic dynamic-programming problem, which is extremely difficult to solve. A suboptimal strategy is proposed, which relies ona normal approximation of the future posterior of θ, independent of future observations. The design criterion obtained involves several mathematical expectations, which are approximated by Laplace method. Finally, stochastic approximation algorithms are also suggested to determine (sub)optimal sequential experiments without having to compute expectations.

How to cite

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Roland Gautier, and Luc Pronzato. "Adaptive control for sequential design." Discussiones Mathematicae Probability and Statistics 20.1 (2000): 97-114. <http://eudml.org/doc/287652>.

@article{RolandGautier2000,
abstract = {The optimal experiment for estimating the parameters of a nonlinear regression model usually depends on the value of these parameters, hence the problem of designing experiments that are robust with respect to parameter uncertainty. Sequential designpermits to adapt the experiment to the value of the parameters, and can thus be considered as a robust design procedure. By designing theexperiments sequentially, one introduces a feedback of information, and thus dynamics, into the design procedure. Several sequential schemes, corresponding to different control policies, are considered. The optimal one corresponds to closed-loop control, and is solution of a stochastic dynamic-programming problem, which is extremely difficult to solve. A suboptimal strategy is proposed, which relies ona normal approximation of the future posterior of θ, independent of future observations. The design criterion obtained involves several mathematical expectations, which are approximated by Laplace method. Finally, stochastic approximation algorithms are also suggested to determine (sub)optimal sequential experiments without having to compute expectations.},
author = {Roland Gautier, Luc Pronzato},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {active control; adaptive control; certainty equivalence; closed-loop control; dynamic programming; open-loop feedback; optimal design; sequential design},
language = {eng},
number = {1},
pages = {97-114},
title = {Adaptive control for sequential design},
url = {http://eudml.org/doc/287652},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Roland Gautier
AU - Luc Pronzato
TI - Adaptive control for sequential design
JO - Discussiones Mathematicae Probability and Statistics
PY - 2000
VL - 20
IS - 1
SP - 97
EP - 114
AB - The optimal experiment for estimating the parameters of a nonlinear regression model usually depends on the value of these parameters, hence the problem of designing experiments that are robust with respect to parameter uncertainty. Sequential designpermits to adapt the experiment to the value of the parameters, and can thus be considered as a robust design procedure. By designing theexperiments sequentially, one introduces a feedback of information, and thus dynamics, into the design procedure. Several sequential schemes, corresponding to different control policies, are considered. The optimal one corresponds to closed-loop control, and is solution of a stochastic dynamic-programming problem, which is extremely difficult to solve. A suboptimal strategy is proposed, which relies ona normal approximation of the future posterior of θ, independent of future observations. The design criterion obtained involves several mathematical expectations, which are approximated by Laplace method. Finally, stochastic approximation algorithms are also suggested to determine (sub)optimal sequential experiments without having to compute expectations.
LA - eng
KW - active control; adaptive control; certainty equivalence; closed-loop control; dynamic programming; open-loop feedback; optimal design; sequential design
UR - http://eudml.org/doc/287652
ER -

References

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  1. [1] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, N. J., 1957. 
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  9. [9] L. Pronzato and E. Walter, Robust experiment design via stochastic approximation, Mathematical Biosciences 75 (1985), 103-120. Zbl0593.62070
  10. [10] L. Pronzato, E. Walter and C. Kulcsár, A dynamical-system approach to sequential design, Model-Oriented Data Analysis III, Proceedings MODA3, St Petersburg, May 1992 (W. G. Müller, H. P. Wynn and A. A. Zhigljavsky, eds.), Physica Verlag, Heidelberg, 11-24. Zbl0860.62061
  11. [11] W. J. Runggaldier, Concepts of optimality in stochastic control, Reliability and Decision (R. Barlow, et al., ed.), Elsevier, Amsterdam, 101-114. 
  12. [12] M. Tanner, Tools for Statistical Inference Methods for Exploration of Posterior Distributions and Likelihood Functions, Springer, Heidelberg 1993. Zbl0777.62003
  13. [13] L. Tierney and J. Kadane, Accurate approximations for posterior moments and marginal densities, Journal of the American Statistical Association 81 (393) (1986), 82-86. Zbl0587.62067
  14. [14] S. Zacks, Problems and approaches in design of experiments for estimation and testing in nonlinear models, Multivariate Analysis IV, (P. Krishnaiah, ed.), North Holland, Amsterdam 1977, 209-223. 

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