混合效应模型的最优区组设计

本文对将驻点个数等于参数个数的混合效应模型做D-最优设计.以两种误差分布的组合为例做近似区组设计,然后给出精确区组设计的设计点位置及区组的权数的解析方程组,给出精确设计比近似设计好的条件.最后,给出数值结果及效率比较.     

Quadrature Methods for Bayesian Optimal Design of Experiments With Nonnormal Prior Distributions

Many optimal experimental designs depend on one or more unknown model parameters. In such cases, it is common to use Bayesian optimal design procedures to seek designs that perform well over an entire prior distribution of the unknown model parameter(s). Generally, Bayesian optimal design procedures are viewed as computationally intensive. This is because they require numerical integration techniques to approximate the Bayesian optimality criterion at hand. The most common numerical integration technique involves pseudo Monte Carlo draws from the prior distribution(s). For a good approximation of the Bayesian optimality criterion, a large number of pseudo Monte Carlo draws is required. This results in long computation times. As an alternative to the pseudo Monte Carlo approach, we propose using computationally efficient Gaussian quadrature techniques. Since, for normal prior distributions, suitable quadrature techniques have already been used in the context of optimal experimental design, we focus on quadrature techniques for nonnormal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients, and any other parameters that are strictly positive or have upper and lower bounds. In this article, we demonstrate the added value of the quadrature techniques we advocate by means of the Bayesian D-optimality criterion in the context of split-plot experiments, but we want to stress that the techniques can be applied to other optimality criteria and other types of experimental designs as well. Supplementary materials for this article are available online.     

A simplified version of strong connectivity in geometries

If there are less than b distinct blocks in a BIB design with b blocks then we say the design has repeated blocks. The set of distinct blocks of a design is called the support of the design. BIB designs with repeated blocks, besides being optimal, have special applications in the design of experiments and controlled samplings. Construction of BIB(ν, b, r, k, λ) designs with repeated blocks becomes complicated whenever the three parameters b, r, and λ are relatively prime. BIB(8, 56, 21, 3, 6) designs are examples of such designs with the smallest number of varieties. BIB(10, 30, 9, 3, 2) designs are such designs with the smallest number of blocks. We make an interesting observation about BIB(8, 56, 21, 3, 6) designs and give a table of such designs with 30 different support sizes. We prove, by construction, that a BIB(10, 30, 9, 3, 2) design exists if and only if the support size belongs to {21, 23, 24, 25, 26, 27, 28, 29, 30}. Other results are also given.     

Optimum mechanism for breaking the confounding effects of mixed-level designs

Fractional factorial designs (FFD’s) are no doubt the most widely used designs in the experimental investigations due to their efficient use of experimental runs to study many factors simultaneously. One consequence of using FFD’s is the aliasing of factorial effects. Follow-up experiments may be needed to break the confounding. A simple strategy is to add a foldover of the initial design, the new fraction is called a foldover design. Combining a foldover design with the original design converts a design of resolution r into a combined design of resolution \(r+1\). In this paper, we take the centered \(L_2\)-discrepancy \(({\mathcal {CD}})\) as the optimality measure to construct the optimal combined design and take asymmetrical factorials with mixed two and three levels, which are most commonly used in practice, as the original designs. New and efficient analytical expressions based on the row distance of the \({\mathcal {CD}}\) for combined designs are obtained. Based on these new formulations, we present new and efficient lower bounds of the \({\mathcal {CD}}\). Using the new formulations and lower bounds as the benchmarks, we may implement a new algorithm for constructing optimal mixed-level combined designs. By this search heuristic, we may obtain mixed-level combined designs with low discrepancy.     

A-optimal designs for an additive cubic model

Chan et al. (1998a) obtained A-optimal designs for an additive quadratic mixture model for q≥3 mixture components. In this paper, we obtain the A-optimal designs for an additive cubic model for q≥3 mixture components using the class of symmetric weighted centroid designs based on barycentres of various depths. We observe that barycentres of depths 0 and 2 are possible support points for an A-optimal design. We have also given the optimal weights of A-optimal designs for 3≤q≤17.     

Bayesian accelerated life test plans for series systems with Weibull component lifetimes

This article presents optimal Bayesian accelerated life test plans for series systems under Type-I censoring scheme. First, the component lifetimes are assumed to follow independent Weibull distributions. The scale parameters of Weibull lifetime distributions are related to the external stress variable through a general stress translation function. For a fixed number of design points, optimal Bayesian ALT plans are first obtained by solving constrained optimization problems under two different Bayesian design criteria. The global optimality of the resulting fixed-point optimal designs is then verified via the General Equivalence Theorem. This article also provides the optimized compromise ALT plans which are extremely useful in real-life applications. A detailed sensitivity analysis is then performed to find out the effect of various planning inputs on the resulting optimal Bayesian ALT plans. A simulation study is then conducted to visualize the resulting sampling variations from the optimal Bayesian ALT plans. Finally, this article considers a series system with dependent component lifetimes. Optimal ALT plans are obtained assuming a Gamma frailty model.     

D-Optimal Designs for Trigonometric Regression Models on a Partial Circle

In the common trigonometric regression model we investigate the D-optimal design problem, where the design space is a partial circle. It is demonstrated that the structure of the optimal design depends only on the length of the design space and that the support points (and weights) are analytic functions of this parameter. By means of a Taylor expansion we provide a recursive algorithm such that the D-optimal designs for Fourier regression models on a partial circle can be determined in all cases. In the linear and quadratic case the D-optimal design can be determined explicitly.     

Robustness of Bayesian D-optimal design for the logistic mixed model against misspecification of autocorrelation

In medicine and health sciences mixed effects models are often used to study time-structured data. Optimal designs for such studies have been shown useful to improve the precision of the estimators of the parameters. However, optimal designs for such studies are often derived under the assumption of a zero autocorrelation between the errors, especially for binary data. Ignoring or misspecifying the autocorrelation in the design stage can result in loss of efficiency. This paper addresses robustness of Bayesian D-optimal designs for the logistic mixed effects model for longitudinal data with a linear or quadratic time effect against incorrect specification of the autocorrelation. To find the Bayesian D-optimal allocations of time points for different values of the autocorrelation, under different priors for the fixed effects and different covariance structures of the random effects, a scalar function of the approximate variance–covariance matrix of the fixed effects is optimized. Two approximations are compared; one based on a first order penalized quasi likelihood (PQL1) and one based on an extended version of the generalized estimating equations (GEE). The results show that Bayesian D-optimal allocations of time points are robust against misspecification of the autocorrelation and are approximately equally spaced. Moreover, PQL1 and extended GEE give essentially the same Bayesian D-optimal allocation of time points for a given subject-to-measurement cost ratio. Furthermore, Bayesian optimal designs are hardly affected either by the choice of a covariance structure or by the choice of a prior distribution.     

Minimax designs for linear regression models with bias in a reproducing kernel Hilbert space in a discrete set

Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the model bias comes from a reproducing kernel Hilbert space. Two different design criteria are proposed by applying the minimax approach for estimating the parameters of the regression response and extrapolating the regression response to points outside of the design space. A simulated annealing algorithm is applied to construct the minimax designs.These minimax designs are compared with the classical D-optimal designs and all-bias extrapolation designs. Numerical results indicate that the simulated annealing algorithm is feasible and the minimax designs are robust against bias caused by model misspecification.     

Local c- and E-optimal Designs for Exponential Regression Models

In this paper we investigate local E- and c-optimal designs for exponential regression models of the form . We establish a numerical method for the construction of efficient and local optimal designs, which is based on two results. First, we consider for fixed k the limit μ _i → γ (i = 1, ... , k) and show that the optimal designs converge weakly to the optimal designs in a heteroscedastic polynomial regression model. It is then demonstrated that in this model the optimal designs can be easily determined by standard numerical software. Secondly, it is proved that the support points and weights of the local optimal designs in the exponential regression model are analytic functions of the nonlinear parameters μ ₁, ... , μ _k . This result is used for the numerical calculation of the local E-optimal designs by means of a Taylor expansion for any vector (μ ₁, ... , μ _k ). It is also demonstrated that in the models under consideration E-optimal designs are usually more efficient for estimating individual parameters than D-optimal designs.     

The usefulness of Bayesian optimal designs for discrete choice experiments

Recently, the use of Bayesian optimal designs for discrete choice experiments, also called stated choice experiments or conjoint choice experiments, has gained much attention, stimulating the development of Bayesian choice design algorithms. Characteristic for the Bayesian design strategy is that it incorporates the available information about people's preferences for various product attributes in the choice design. This is in contrast with the linear design methodology, which is also used in discrete choice design and which depends for any claims of optimality on the unrealistic assumption that people have no preference for any of the attribute levels. Although linear design principles have often been used to construct discrete choice experiments, we show using an extensive case study that the resulting utility‐neutral optimal designs are not competitive with Bayesian optimal designs for estimation purposes. Copyright © 2011 John Wiley & Sons, Ltd.     

Multimodel control approach for electrohydraulic servo systems

The paper proposes a control design approach based on a multipoint linearization method for linear electro-hydraulic servo systems. The nonlinear model of servo system is linearized around of operational points and Matlab environment is used to adjust the points distribution based on an errors estimation. These models are used to design an optimal controller which online modifies the feedback control parameters. Simulation and experimental results are provided to show the effectiveness of approach. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of optimal supersaturated designs by the packing method

A supersaturated design is essentially a factorial design with the equal occurrence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently unrelated kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties.     

Optimal design for smoothing splines

In the common nonparametric regression model we consider the problem of constructing optimal designs, if the unknown curve is estimated by a smoothing spline. A special basis for the space of natural splines is introduced and the local minimax property for these splines is used to derive two optimality criteria for the construction of optimal designs. The first criterion determines the design for a most precise estimation of the coefficients in the spline representation and corresponds to D-optimality, while the second criterion is the G-optimality criterion and corresponds to an accurate prediction of the curve. Several properties of the optimal designs are derived. In general, D- and G-optimal designs are not equivalent. Optimal designs are determined numerically and compared with the uniform design.     

A bi-objective turning restriction design problem in urban road networks

This paper introduces a bi-objective turning restriction design problem (BOTRDP), which aims to simultaneously improve network traffic efficiency and reduce environmental pollution by implementing turning restrictions at selected intersections. A bi-level programming model is proposed to formulate the BOTRDP. The upper level problem aims to minimize both the total system travel time (TSTT) and the cost of total vehicle emissions (CTVE) from the viewpoint of traffic managers, and the lower level problem depicts travelers’ route choice behavior based on stochastic user equilibrium (SUE) theory. The modified artificial bee colony (ABC) heuristic is developed to find Pareto optimal turning restriction strategies. Different from the traditional ABC heuristic, crossover operators are captured to enhance the performance of the heuristic. The computational experiments show that incorporating crossover operators into the ABC heuristic can indeed improve its performance and that the proposed heuristic significantly outperforms the non-dominated sorting genetic algorithm (NSGA) even if different operators are randomly chosen and used in the NSGA as in our proposed heuristic. The results also illustrate that a Pareto optimal turning restriction strategy can obviously reduce the TSTT and the CTVE when compared with those without implementing the strategy, and that the number of Pareto optimal turning restriction designs is smaller when the network is more congested but greater network efficiency and air quality improvement can be achieved. The results also demonstrate that traffic information provision does have an impact on the number of Pareto optimal turning restriction designs. These results should have important implications on traffic management.     

基于M-CVaR和生产资金约束的供应链协调研究

本文针对供应商面临生产资金约束的情况,在需求随机条件下研究了由一个风险中性零售商和一个具有风险偏好的供应商组成的二级供应链的协调问题。文章在零售商享有批发价折扣的提前支付和供应商银行信贷两种融资方式下,分别建立了零售商的最优决策模型以及基于M-CVaR测度工具的供应链整体订购决策模型,并给出了采用两种不同融资模式时零售商和供应链整体的最优订购量以及供应链的协调条件,分析了供应商的风险偏好对供应链整体最优决策及协调条件的影响。最后通过算例验证了文章的主要结论。研究表明,当提前支付价格折扣大于临界值时,零售商会选择提前支付货款;提前支付模式下供应链整体的最优订购量大于银行信贷模式;随着供应商的风险偏好由风险规避向风险喜好转变,供应链整体的最优订购量逐渐增大。     

混料切片试验设计

混料试验设计在众多领域中都有广泛的应用,有时试验者不仅仅需要考虑各混料成分所占比例对响应变量的影响,同时还关心其它被称为过程变量的因素.在实际中,对于这类问题通常使用的设计方案是混料设计和因子设计的组合设计.这种组合设计在过程变量的不同水平组合下,使用的是相同的设计阵,因此空间填充性较差.基于混料球体堆积设计,文章提出了一类新的混料设计,称之为混料切片设计,它的整体设计和所有子设计(过程变量的每一水平组合对应的混料设计)都具有很好的空间填充性,从而比组合设计有更好的模型稳健性.基于同余子群的陪集分解方法,针对过程变量水平组合数的不同情况提出了相应的简单快速的构造算法,文章最后的数值例子解释了算法的可行性和设计的有效性.     

An upper bound of the number of tests in pooling designs for the error-tolerant complex model

Recently pooling designs have been used in screening experiments in molecular biology. In some applications, the property to be screened is defined on subsets of items, instead of on individual items. Such a model is usually referred to as the complex model. In this paper we give an upper bound of the number of tests required in a pooling design for the complex model (with given design parameters) where experimental errors are allowed.     

Construction of efficient experimental designs under multiple resource constraints

The aim of this paper is twofold. First, we introduce ‘resource constraints’ as a general concept that covers many practical restrictions on experimental design. Second, to compute optimal or near‐optimal exact designs of experiments under multiple resource constraints, we propose a tabu search heuristic related to the Detmax procedure. To illustrate the scope and performance of the proposed method, we chose the criterion of D‐optimality and computed efficient designs for (i) a block model with limits on the numbers of blocks and on the replications of treatments, (ii) a quadratic regression model with simultaneous marginal and cost constraints, and (iii) a non‐linear regression model with simultaneous direct and cost constraints. As we show, the proposed method generates similar or better results compared with algorithms specialized for computing optimal designs under less general constraints. Copyright © 2015 John Wiley & Sons, Ltd.     

Learning horizon and optimal alliance formation

We develop a theoretical Bayesian learning model to examine how a firm’s learning horizon, defined as the maximum distance in a network of alliances across which the firm learns from other firms, conditions its optimal number of direct alliance partners under technological uncertainty. We compare theoretical optima for a ‘close’ learning horizon, where a firm learns only from direct alliance partners, and a ‘distant’ learning horizon, where a firm learns both from direct and indirect alliance partners. Our theory implies that in high tech industries, a distant learning horizon allows a firm to substitute indirect for direct partners, while in low tech industries indirect partners complement direct partners. Moreover, in high tech industries, optimal alliance formation is less sensitive to changes in structural model parameters when a firm’s learning horizon is distant rather than close. Our contribution lies in offering a formal theory of the role of indirect partners in optimal alliance portfolio design that generates normative propositions amenable to future empirical refutation.