Experimental design for estimating the optimum point in a response surface

The problem of optimal experimental design for response optimization is considered. The optimal point (control)x* of a response surface is to be determined by estimating the response parametersθ from measurements performed at design pointsxi,i=1,...,N. Classical sequential approaches for choosing thexi's are recalled. A loss function related to the issue of response optimization is used to define control-oriented design criteria. The design policies differ depending on whether least-squares or minimum risk estimation is used to estimateθ. Connections between various criteria suggested in the literature are exhibited. Special attention is given to quadratic model responses. Most approaches presented assume that the response is correctly described by a given parametric function over the region of interest. Possible deterministic departures from this function raise the problem of model robustness, and the literature on the subject is briefly surveyed.

[1]  Jerome Sacks,et al.  ROBUST DESIGNS FOR REGRESSION PROBLEMS**This work was partially supported by grants from the National Science Foundation. , 1977 .

[2]  D. Steinberg.  Model robust response surface designs: Scaling two-level factorials , 1985 .

[3]  E. Walter,et al.  How to Design Experiments that are Robust to Parameter Uncertainty , 1985 .

[4]  Jerome Sacks,et al.  Designs for Computer Experiments , 1989 .

[5]  D. Lindley.  The Choice of Variables in Multiple Regression , 1968 .

[6]  Jürgen Pilz,et al.  Once more: optimal experimental design for regression models (with discussion) , 1987 .

[7]  Y. Bar-Shalom,et al.  Dual effect, certainty equivalence, and separation in stochastic control , 1974 .

[8]  W. G. Hunter,et al.  Experimental Design: Review and Comment , 1984 .

[9]  Jerome Sacks,et al.  LINEAR ESTIMATION FOR APPROXIMATELY LINEAR MODELS , 1978 .

[10]  D. Lindley.  On a Measure of the Information Provided by an Experiment , 1956 .

[11]  A. R. Manson,et al.  Minimum Bias Estimation and Experimental Design for Response Surfaces , 1969 .

[12]  J. Kiefer.  Optimum Designs for Fitting Biased Multiresponse Surfaces , 1973 .

[13]  Stephen M. Stigler,et al.  Optimal Experimental Design for Polynomial Regression , 1971 .

[14]  J. Buonaccorsi,et al.  OPTIMAL DESIGNS FOR RATIOS OF LINEAR-COMBINATIONS IN THE GENERAL LINEAR-MODEL , 1986 .

[15]  Changbao Wu,et al.  Asymptotic inference from sequential design in a nonlinear situation , 1985 .

[16]  Jerome Sacks,et al.  Some Model Robust Designs in Regression , 1984 .

[17]  K. Chaloner.  Optimal Bayesian Experimental Design for Linear Models , 1984 .

[18]  E. Walter,et al.  Robust experiment design via maximin optimization , 1988 .

[19]  Eric Walter,et al.  Qualitative and quantitative experiment design for phenomenological models - A survey , 1990, Autom..

[20]  M. Goldstein.  The linear Bayes regression estimator under weak prior assumptions , 1980 .

[21]  S. Silvey,et al.  A sequentially constructed design for estimating a nonlinear parametric function , 1980 .

[22]  B. Blight,et al.  A Bayesian approach to model inadequacy for polynomial regression , 1975 .

[23]  H. Chernoff.  Approaches in Sequential Design of Experiments , 1973 .

[24]  S. K. Chatterjee,et al.  Response Surface Designs for Estimating the Optimal Point , 1981 .

[25]  R. J. Brooks.  Optimal regression design for control in linear regression , 1977 .

[26]  Corwin L. Atwood.  Convergent Design Sequences, for Sufficiently Regular Optimality Criteria, II: Singular Case , 1980 .

[27]  Henry P. Wynn,et al.  Maximum entropy sampling , 1987 .

[28]  Jürgen Pilz,et al.  Bayesian estimation and experimental design in linear regression models , 1992 .

[29]  D. Titterington,et al.  Inference and sequential design , 1985 .

[30]  H. Scheffé.  Experiments with Mixtures , 1958 .

[31]  K. Chaloner,et al.  Optimal Bayesian design applied to logistic regression experiments , 1989 .

[32]  C. Currin,et al.  A Bayesian Approach to the Design and Analysis of Computer Experiments , 1988 .

[33]  G. Box,et al.  A Basis for the Selection of a Response Surface Design , 1959 .

[34]  W. J. Studden.  Elfving's Theorem and Optimal Designs for Quadratic Loss , 1971 .

[35]  Alessandra Giovagnoli,et al.  Bayes D-optimal and E-optimal block designs , 1983 .

[36]  J. Kiefer,et al.  Stochastic Estimation of the Maximum of a Regression Function , 1952 .

[37]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[38]  L. Pesotchinsky,et al.  Optimal Robust Designs: Linear Regression in $R^k$ , 1982 .

[39]  D. Naidu,et al.  Optimal Control Systems , 2018 .

[40]  S. Zacks,et al.  Problems and Approaches in Design of Experiments for Estimation and Testing in Non-Linear Models. , 1975 .

[41]  William J. Hill,et al.  A Review of Response Surface Methodology: A Literature Survey* , 1966 .

[42]  A linear dynamic feedback controller for stochastic systems with unknown parameters , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[43]  W. J. Studden,et al.  Theory Of Optimal Experiments , 1972 .

[44]  S. D. Silvey,et al.  Optimal design measures with singular information matrices , 1978 .

[45]  L. Pronzato,et al.  ROBUST EXPERIMENT DESIGN: BETWEEN QUALITATIVE AND QUANTITATIVE IDENTIFIABILITIES , 1987 .

[46]  D-optimal designs for estimating the optimum point in a quadratic response surface - rectangular region , 1989 .

[47]  E. Walter,et al.  Robust experiment design via stochastic approximation , 1985 .

[48]  G. Elfving.  Optimum Allocation in Linear Regression Theory , 1952 .

[49]  Andrej Pázman,et al.  Foundations of Optimum Experimental Design , 1986 .

[50]  J. S. Hunter,et al.  Multi-Factor Experimental Designs for Exploring Response Surfaces , 1957 .

[51]  Peter D. H. Hill,et al.  A Review of Experimental Design Procedures for Regression Model Discrimination , 1978 .

[52]  A. Dvoretzky.  On Stochastic Approximation , 1956 .

[53]  J. Kiefer.  General Equivalence Theory for Optimum Designs (Approximate Theory) , 1974 .

[54]  W. Welch.  A mean squared error criterion for the design of experiments , 1983 .

[55]  E. Ziegel.  Optimal design and analysis of experiments , 1990 .

[56]  Samuel H. Brooks,et al.  Optimum Estimation of Gradient Direction in Steepest Ascent Experiments , 1961 .

[57]  W. J. Studden.  Some Robust-Type D-Optimal Designs in Polynomial Regression , 1982 .

[58]  Z. Galil,et al.  Comparison of Simplex Designs for Quadratic Mixture Models , 1977 .

[59]  Donald Ylvisaker,et al.  Prediction and Design , 1987 .