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multiplicative algorithm for discriminating between Arrhenius and non-Arrhenius behaviour

CHEMOMETRICS AND INTELLIGENT LABORATORY SYSTEMS 2014年 139卷 146-155页

作者： Martin-Martin, Raul Dorta-Guerra, Roberto Torsney, Ben Sch Architecture E-45071 Toledo Spain Dept Math Stat & Operat Res E-38271 San Cristobal la Laguna Spain Univ Glasgow Sch Math & Stat Glasgow G12 8QW Lanark Scotland

The Arrhenius equation k(A) = Ae(-E/Rt) has found numerous applications throughout chemical kinetics for diverse rate processes. This equation involves the assumption that the pre-exponential factor, A, does not vary with temperature. For simple reactions, deviations from this equation are usually quite small, and in only a few instances are they at all readily detectable. However kinetic measurements over wide ranges of temperature show up the inadequacy of the Arrhenius expression (Smith, 2008 [1]). The temperature dependence of the linear parameter is widely used as an alternative model. Consequently, a correct selection of the model and a correct estimation of the parameters are crucial tasks. The focus of this paper is the construction of a new procedure based on the multiplicative algorithm in order to determining optimal experimental conditions for discriminating between two rival models. Even for moderate examples the calculation of T-optimal designs is not straightforward. There are no known specific iterative numerical techniques for constructing T-optimum designs;there is just the classical adaptation of the Wynn-Fedorov scheme (Atkinson and Fedorov, 1975;Fedorov and Hackl, 1997), which is far from being satisfactory to solve these computational problems. The results are illustrated by numerical examples for different deviations of the Arrhenius equation. On the other hand we demonstrate in several examples that the new algorithm is more efficient than those existing in the literature. (C) 2014 Elsevier B.V. All rights reserved.

关键词： T-optimum design Arrhenius equation Modified Arrhenius equation multiplicative algorithm

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Finding optimal design in nonlinear mixed effect models using multiplicative algorithms

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COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2021年 207卷 106126-106126页

作者： Seurat, Jeremy Tang, Yuxin Mentre, France Nguyen, Thu Thuy Univ Paris INSERM IAME F-75006 Paris France

Background and objectives: To optimize designs for longitudinal studies analyzed by nonlinear mixed effect models (NLMEMs), the Fisher information matrix (FIM) can be used. In this work, we focused on the multiplicative algorithms, previously applied in standard individual regression, to find optimal designs for ***: We extended multiplicative algorithms to mixed models and implemented the algorithm both in R and in C. Then, we applied the algorithm to find D-optimal designs in two longitudinal data examples, one with continuous and one with binary ***: For these examples, we quantified the improved speed when C is used instead of R. Design optimization using the multiplicative algorithm led to designs with D-efficiency gains between 13% and 25% compared to non-optimized ***: We found that the multiplicative algorithm can be used efficiently to design longitudinal studies.

关键词： Nonlinear mixed effect model Fisher information matrix multiplicative algorithm Optimal design D-optimality Longitudinal studies

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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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multiplicative algorithm for computing D-optimum designs for pVT measurements

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CHEMOMETRICS AND INTELLIGENT LABORATORY SYSTEMS 2012年第1期111卷 20-27页

作者： Martin-Martin, Raul Rodriguez-Aragon, Licesio J. Torsney, Ben Sch Comp Sci E-13071 Ciudad Real Spain Sch Architecture E-45071 Toledo Spain Univ Glasgow Sch Math & Stat Glasgow G12 8QW Lanark Scotland Univ Castilla La Mancha Dept Matemat Inst Matemat Aplicada Ciencia & Ingn E-13071 Ciudad Real Spain

Multifactor models are present in industry and in biological experiments: therefore optimum designs are a valuable tool for experimenters, leading to estimators of the parameters with minimum variance. A generalization of the multiplicative algorithm to find locally D-optimum designs for multifactor models as Tait-like equations is here proposed. The method consists of transforming a conceived set of design points over a finite interval into proportions of the design interval defined by the sub-intervals between successive points. Examples for different modifications of the Tait equation are here presented: a three parameter model under isothermal conditions with pressure as the independent variable and two different modifications of an eight parameter model (Rackett equation) with pressure and temperature as design variables. Optimum design for the first example is equally supported at three points while the two-factor modifications of the Tait equation show designs with more support points than unknown parameters and different weights for each point Efficiencies of real experimental designs are computed and suitable more efficient experimental designs, satisfying experimental demands are also proposed. (C) 2011 Elsevier B.V. All rights reserved.

关键词： D-optimum design Tait equation multiplicative algorithm 2008 MSC: 62K05

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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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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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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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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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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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Combined algorithm to compute D-optimal designs

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2015年 278卷 248-257页

作者： Martin Martin, Raul Camacha Gutierrez, Irene Garcia Univ Castilla La Mancha Dept Math Toledo Spain

algorithmic techniques for computing optimal designs continue being a need in the optimal experimental design field. The increasing interest in finding the optimal experimental conditions makes that new methods are demanded for more complex frameworks showing more realistic situations. Numerical techniques are often the unique viable option for computing these designs since to tackle analytically the problem results impracticable in most of cases. Wynn-Fedorov algorithm, multiplicative algorithm and their modifications are the more frequently used methods in the literature for computing D-optimal designs. However, they are not always suitable and efficient to compute optimal designs since their procedures are very limited by the computational requirements. A new algorithm to obtain D-optimal designs is proposed in this paper. It is based on combining suitable strategies followed by the traditional algorithms. A proof of the convergence is provided and several numerical examples are presented to illustrate its improved results. (C) 2014 Elsevier B.V. All rights reserved.

关键词： D-optimum design multiplicative algorithm Wynn-Fedorov algorithm Combined algorithm

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