D-optimal designs for estimating the optimum point in a quadratic response surface - rectangular region
01 Oct 1989-Journal of Statistical Planning and Inference (North-Holland)-Vol. 23, Iss: 2, pp 243-252
TL;DR: In this article, the problem of estimating the optimum factor combinations in a quantitative multifactor experiment under the assumption that the response function is quadratic concave, solutions are obtained for the rectangular experimental region using the criterion of extended D-optimality.
Abstract: The problem considered is that of estimating the optimum factor combinations in a quantitative multifactor experiment Under the assumption that the response function is quadratic concave, solutions are obtained for the rectangular experimental region using the criterion of extended D-optimality Some of the results are developed algebraically and others numerically A method of deriving optimal designs is also indicated
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TL;DR: In this paper, the authors consider designs when interest is in estimating the optimal factor combination in a multiple quadratic regression setup, supposing that this factor combination belongs to a given set.
Abstract: We consider designs when interest is in estimating the optimal factor combination in a multiple quadratic regression setup, supposing that this factor combination belongs to a given set. By involving the concepts of admissibility and invariance of designs we substantially reduce the problem of calculating minimax designs. Exemplary, we give optimal designs for some setups on the ball and on the cube.
25 citations
TL;DR: The problem of optimal experimental design for response optimization is considered in this paper, where 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.
Abstract: 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.
22 citations
Cites background or methods from "D-optimal designs for estimating th..."
...The extended D-optimality criterion [ 33 ] is given by...
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...An analytical expression for this criterion can be obtained and optimal designs are computed by Mandal [ 33 ], assuming...
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...Concerning the optimization of (34), except some cases where an analytical expression of the criterion can be obtained (see [ 33 ] and Section 6), there seems to be no alternative to evaluating expectation at each step of the optimization algorithm....
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...We shall see in Section 6 that the criterion (34) coincides with the extended D-optimality criterion of Mandal [ 33 ] defined in the context of quadratic model responses....
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TL;DR: In this article, the authors provide a compilation of theoretic response surface designs (RSDs) for process or product optimization studies to explore the input-response relationship, and provide a theoretical analysis of the relationship between RSDs.
Abstract: Response Surface Designs (RSDs) are widely used in process or product optimization studies to explore the input-response relationship. This paper is an attempt to provide a compilation of theoretic...
6 citations
Cites methods from "D-optimal designs for estimating th..."
...Afterwards, several workers contributed in this area for the multifactor case using different optimality criteria (see e.g., Chatterjee and Mandal 1981, 1985; Mandal 1982, 1986, 1989; Mandal and Heiligers 1992; Mukhopadhyay and Sinha 1998; Fedorov and M€uller 1997 and others)....
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TL;DR: In this paper, a central composite design which maximizes both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters is presented.
Abstract: Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.
5 citations
TL;DR: In this article, an attempt has been made to find optimum designs when the experimental region is a simplex or is cuboidal inside the simplex and does not contain the extreme points.
Abstract: In a mixture experiment, the response depends on the mixing proportions of the components present in the mixture. Optimum designs are available for the estimation of parameters of the models proposed in such situations. However, these designs are found to include the vertex points of the simplex Ξ defining the experimental region, which are not mixtures in the true sense. Recently, Mandal et al. (2015) derived optimum designs when the experiment is confined to an ellipsoidal region within Ξ, which does not include the vertices of Ξ. In this paper, an attempt has been made to find optimum designs when the experimental region is a simplex or is cuboidal inside Ξ and does not contain the extreme points.
4 citations
Cites methods from "D-optimal designs for estimating th..."
...From Mandal (1989), we have the following result for D-optimal design to estimate the parameters of the model (2.29): Theorem....
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References
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TL;DR: The work described in this article is the result of a study extending over the past few years by a chemist and a statistician, which has come about mainly in answer to problems of determining optimum conditions in chemical investigations, but they believe that the methods will be of value in other fields where experimentation is sequential and the error fairly small.
Abstract: The work described is the result of a study extending over the past few years by a chemist and a statistician. Development has come about mainly in answer to problems of determining optimum conditions in chemical investigations, but we believe that the methods will be of value in other fields where experimentation is sequential and the error fairly small.
4,359 citations
TL;DR: For general optimality criteria, this article obtained criteria equivalent to $\Phi$-optimality under various conditions on ''Phi'' and showed that such equivalent criteria are useful for analytic or machine computation of ''phi''-optimum designs.
Abstract: For general optimality criteria $\Phi$, criteria equivalent to $\Phi$-optimality are obtained under various conditions on $\Phi$. Such equivalent criteria are useful for analytic or machine computation of $\Phi$-optimum designs. The theory includes that previously developed in the case of $D$-optimality (Kiefer-Wolfowitz) and $L$-optimality (Karlin-Studden-Fedorov), as well as $E$-optimality and criteria arising in response surface fitting and minimax extrapolation. Multiresponse settings and models with variable covariance and cost structure are included. Methods for verifying the conditions required on $\Phi$, and for computing the equivalent criteria, are illustrated.
736 citations
TL;DR: In this article, the problem of multilinear regression on the simplex has been studied and a sufficient condition for optimality is given, and a corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality.
Abstract: This paper consists of new results continuing the series of papers on optimal design theory by Kiefer (1959), (1960), (1961), Kiefer and Wolfowitz (1959), (1960), Farrell, Kiefer and Walbran (1965) and Karlin and Studden (1966a). After disposing of the necessary preliminaries in Section 1, we show in Section 2 that in several classes of problems an optimal design for estimating all the parameters is supported only on certain points of symmetry. This is applied to the problem (introduced by Scheffe (1958)) of multilinear regression on the simplex. In Section 3 we consider optimality when nuisance parameters are present. A new sufficient condition for optimality is given. A corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality, and we prove the natural invariance theorem involving this condition. These results are applied to the problem of multilinear regression on the simplex when estimating only some of the parameters. Section 4 consists primarily of a number of bounds on the efficiency of designs; these are summarized at the beginning of that section.
186 citations
23 citations
TL;DR: In this paper, the problem of estimating the optimum factor combinations in a multifactor, multiresponse experiment has been considered, the criterion used is the extended D-optimality criterion.
Abstract: The problem of estimating the optimum factor combinations in a multifactor, multiresponse experiment has been considered, The criterion used is the extended D-optimality criterion. The property of convexity of a transform of the criterion function and the principle of invariance have been used to find optimum designs. This is a generalization of the problem considered by Mandai (1982).
9 citations