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Approximate D-optimal designs of experiments on the convex hull of a finite set of information matrices

MATHEMATICA SLOVACA 2009年第6期59卷 693-704页

作者： Harman, Radoslav Trnovska, Maria Comenius Univ Fac Math Phys & Informat Dept Appl Math & Stat SK-84248 Bratislava Slovakia

In the paper we solve the problem of D (a"<)-optimal design on a discrete experimental domain, which is formally equivalent to maximizing determinant on the convex hull of a finite set of positive semidefinite matrices. The problem of D (a"<)-optimality covers many special design settings, e.g., the D-optimal experimental design for multivariate regression models. For D (a"<)-optimal designs we prove several theorems generalizing known properties of standard D-optimality. Moreover, we show that D (a"<)-optimal designs can be numerically computed using a multiplicative algorithm, for which we give a proof of convergence. We illustrate the results on the problem of D-optimal augmentation of independent regression trials for the quadratic model on a rectangular grid of points in the plane.

关键词： D-optimal design multivariate regression multiplicative algorithm D-optimal augmentation of trials

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Optimal design measures under asymmetric errors, with application to binary design points

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2015年 159卷 28-36页

作者： Bose, Mausumi Mukerjee, Rahul Indian Stat Inst Appl Stat Div Kolkata 700108 India Indian Inst Management Calcutta Kolkata 700104 India

We study the optimal design problem under second-order least squares estimation which is known to outperform ordinary least squares estimation when the error distribution is asymmetric. First, a general approximate theory is developed, taking due cognizance of the nonlinearity of the underlying information matrix in the design measure. This yields necessary and sufficient conditions that a D- or A-optimal design measure must satisfy. The results are then applied to find optimal design measures when the design points are binary. The issue of reducing the support size of the optimal design measure is also addressed. (C) 2014 Elsevier B.V. All rights reserved.

关键词： A-criterion D-criterion Directional derivative multiplicative algorithm Second-order least squares Support size Weighing design

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Quasi-Newton algorithm for optimal approximate linear regression design: Optimization in matrix

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2019年 198卷 62-78页

作者： Gaffke, N. Schwabe, R. Univ Magdeburg Fac Math PF 4120 D-39016 Magdeburg Germany

Given a linear regression model and an experimental region for the independent variable, the problem of finding an optimal approximate design calls for minimizing a convex optimality criterion over a convex set of information matrices of feasible approximate designs. For numerical solution pure gradient methods are often used by design theorists, as vertex direction, vertex exchange, multiplicative algorithms, or combinations hereof. These methods have two major deficiencies: a slow convergence rate after a quick but rough approximation to the optimum, and often a large support of the obtained nearly optimal design. The latter feature is related to the fact that the methods optimize in the space of design measures which is usually of high or even infinite dimension, whereas the dimension of the information matrices is often small or moderate. For such situations a quasi-Newton method is revisited which was originally established by Gaffke & Heiligers (1996). In the present paper new possibilities of its application are demonstrated. The algorithm optimizes in matrix space. It shows a good global and an excellent local convergence behavior resulting in an accurate approximation of the optimum. A crucial subroutine solves convex quadratic minimization over the set of information matrices via repeated linear minimization over that set, providing thus the quasi-Newton step of the algorithm. This may also be of interest as a tool for computing an approximate design from a given information matrix and such that the support size of the design keeps Caratheodory's bound. Illustrations are given for D- and I-optimality in particular multivariate random coefficient regression models and for T-optimal discriminating design in univariate polynomial models. Moreover, the behavior of the algorithm is tested for cases of larger dimensions: D- and 1-optimal design for a third order polynomial model in several variables on a discretized cube. (C) 2018 Elsevier B.V. All rights reserv

关键词： Information matrix Optimality criterion Local optimal design Convex hull Efficiency Gradient Local quadratic approximation Convex quadratic minimization BEGS update multiplicative algorithm

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algorithmic and analytical construction of efficient designs in small blocks for comparing consecutive pairs of treatments

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JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 2017年第16期87卷 3195-3207页

作者： Huda, S. Mukerjee, Rahul Kuwait Univ Dept Stat & OR Fac Sci POB 5969 Safat 13060 Kuwait Indian Inst Management Calcutta Kolkata India

Optimal block designs in small blocks are explored under the A-, E- and D-criteria when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first formulate the problem via approximate theory which leads to a convenient multiplicative algorithm for obtaining A-optimal design measures. This, in turn, yields highly efficient exact designs, under the A-criterion, even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available in possibly several stages. Illustrative examples are given and tables of A-optimal design measures are provided. Approximate theory is also seen to yield analytical results on E- and D-optimal design measures.

关键词： A-criterion approximate theory D-criterion E-criterion exact design multiplicative algorithm nesting nonbinary block

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An efficient algorithm for Elastic I-optimal design of generalized linear models

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CANADIAN JOURNAL OF STATISTICS-REVUE CANADIENNE DE STATISTIQUE 2021年第2期49卷 438-470页

作者： Li, Yiou Deng, Xinwei Depaul Univ Dept Math Sci Chicago IL 60604 USA Virginia Tech Dept Stat Blacksburg VA 24061 USA

The generalized linear models (GLMs) are widely used in statistical analysis and the related design issues are undoubtedly challenging. The state-of-the-art works mostly apply to design criteria on the estimates of regression coefficients. The prediction accuracy is usually critical in modern decision-making and artificial intelligence applications. It is of importance to study optimal designs from the prediction aspects for GLMs. In this work, we consider Elastic I-optimality as a prediction-oriented design criterion for GLMs, and develop an efficient algorithm for such EI-optimal designs. By investigating theoretical properties for the optimal weights of any set of design points and extending the general equivalence theorem to the EI-optimality for GLMs, the proposed efficient algorithm adequately combines the Fedorov-Wynn algorithm and the multiplicative algorithm. It achieves great computational efficiency with guaranteed convergence. Numerical examples are conducted to evaluate the feasibility and computational efficiency of the proposed algorithm.

关键词： Fast convergence multiplicative algorithm prediction-oriented sequential algorithm

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Optimal designs with string property under asymmetric errors and SLS estimation

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STATISTICAL PAPERS 2018年第3期59卷 1255-1268页

作者： Huda, S. Mukerjee, Rahul Kuwait Univ Fac Sci Dept Stat & OR POB 5969 Safat 13060 Kuwait Indian Inst Management Calcutta Joka Diamond Harbour Rd Kolkata 700104 India

We consider the optimal design problem when the design space consists of binary vectors with a string property, i.e., a single stretch of ones. This is done in the framework of second-order least squares estimation which is known to outperform ordinary least squares estimation when the error distribution is asymmetric. Analytical as well as computational results on optimal design measures, under the D- and A-criteria, are obtained. The issue of robustness to the unknown skewness parameter of the error distribution is also explored. Finally, we present several procedures which entail N-run designs that are highly efficient, if not optimal.

关键词： Design measure multiplicative algorithm N-run design Robustness Uniform measure

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MONOTONIC CONVERGENCE OF A GENERAL algorithm FOR COMPUTING OPTIMAL DESIGNS

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ANNALS OF STATISTICS 2010年第3期38卷 1593-1606页

作者： Yu, Yaming Univ Calif Irvine Dept Stat Irvine CA 92697 USA

Monotonic convergence is established for a general class of multiplicative algorithms introduced by Silvey, Titterington and Torsney [Comm. Statist. Theory Methods 14 (1978) 1379-1389] or computing optimal designs. A conjecture of Titterington [Appl. Stat. 27(1978) 227-234] is confirmed as a consequence. Optimal designs for logistic regression are used as an illustration.

关键词： A-optimality auxiliary variables c-optimality D-optimality experimental design generalized linear models multiplicative algorithm

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Optimal designs subject to cost constraints in simultaneous equations models

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TEST 2015年第4期24卷 701-713页

作者： Casero-Alonso, Victor Lopez-Fidalgo, Jesus Univ Castilla La Mancha Inst Math Appl Sci & Engn Dept Math E-13071 Ciudad Real Spain

A procedure based on a multiplicative algorithm for computing optimal experimental designs subject to cost constraints in simultaneous equations models is presented. A convex criterion function based on a usual criterion function and an appropriate cost function is considered. A specific L-optimal design problem and a numerical example are taken from Conlisk (J Econ 11:63-76, 1979) to compare the procedure. The problem would need integer nonlinear programming to obtain exact designs. To avoid this, he solves a continuous nonlinear programming problem and then he rounds off the number of replicates of each experiment. The procedure provided in this paper reduces dramatically the computational efforts in computing optimal approximate designs. It is based on a specific formulation of the asymptotic covariance matrix of the full-information maximum likelihood estimators, which simplifies the calculations. The design obtained for estimating the structural parameters of the numerical example by this procedure is not only easier to compute, but also more efficient than the design provided by Conlisk.

关键词： Approximate design Cost constraints Exact design L-optimal design multiplicative algorithm Simultaneous equations Structural equations

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Efficient algorithms for constructing D- and I-optimal exact designs for linear and non-linear models in mixture experiments

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SORT-STATISTICS AND OPERATIONS RESEARCH TRANSACTIONS 2019年第1期43卷 163-189页

作者： Martin Martin, Raul Garcia-Camacha Gutierrez, Irene Torsney, Bernard Univ Castilla La Mancha Dept Math Ciudad Real Spain Univ Glasgow Sch Math & Stat Glasgow Lanark Scotland

The problem of finding optimal exact designs is more challenging than that of approximate optimal designs. In the present paper, we develop two efficient algorithms to numerically construct exact designs for mixture experiments. The first is a novel approach to the well-known multiplicative algorithm based on sets of permutation points, while the second uses genetic algorithms. Using (i) linear and non-linear models, (ii) D- and I-optimality criteria, and (iii) constraints on the ingredients, both approaches are explored through several practical problems arising in the chemical, pharmaceutical and oil industry.

关键词： Optimal experimental design D-optimality I-optimality mixture experiments multiplicative algorithm genetic algorithm exact designs

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A new monotonic algorithm for the E-optimal experiment design problem

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STATISTICS & PROBABILITY LETTERS 2021年 174卷 109097-109097页

作者： Sahu, Nitesh Babu, Prabhu Indian Inst Technol Delhi Ctr Appl Res Elect New Delhi 110016 India

In this paper, we develop a new monotonic algorithm for the E-optimal design problem, for which no simple monotonic algorithm is known to exist, using the idea of majorization-minimization. The available algorithms in the literature have no simple closed update equations whereas the proposed new algorithm has simple closed form update equations. The new algorithm is illustrated through numerical examples, and is shown to be competitive compared with the interior point method and existing state-of-the-art algorithm. (C) 2021 Elsevier B.V. All rights reserved.

关键词： E-optimal design criterion majorization-minimization multiplicative algorithm Experiment design

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