Construction Of Optimal Designs Research Articles

Construction of orthogonal maximin distance designs

Maximin distance designs and orthogonal designs are two attractive classes of space-filling designs for computer experiments, but their theoretical constructions are challenging, especially the construction of optimal designs in terms of both the maximin distance and orthogonality criteria. This paper presents a systematic method for constructing orthogonal maximin distance designs with flexible numbers of runs and factors. The method is carried out by rotating the subarrays of a saturated two-level regular design in Yates order or its circular shifting version. The principal objective is to construct high-level designs from two-level designs, and the method is effective because the performance of high-level designs is determined by that of two-level designs under both the maximin distance and orthogonality criteria. The proposed method is also generalized by rotating the subarrays of a saturated two-level nonregular design such that the resulting designs have flexible run sizes. Comparison results reveal that the resulting orthogonal designs are well worthy of recommendation under the maximin distance criterion. An illustrative example is provided to show that the proposed designs have a good two-dimensional stratification property. An application is given to present the effectiveness of the proposed designs in building statistical surrogate models.

On the Difference in Cycling Pattern on Linear and Higher-Order Effect Designs

D-optimality is a design criterion that seeks to maximize the determinant of the information matrix, or equivalently minimize the determinant of the inverse information matrix of the design. This design criterion results in maximizing the differential Shannon information content of the parameter estimates. Cycling, a phenomenal problem associated with the construction of optimal designs, impedes the rate of convergence to such desired optimum, whenever it occurs in a variance exchange process. Different polynomial functions may have varying effects on the pattern of convergence due to cycling. This paper seeks to determine the nature and extent to which the influence of cycling affects the pattern of convergence on Linear, Interactive, and Quadratic order effect designs. The variance exchange algorithmic search method was adopted based on the philosophy of numerically searching the design space for optimum designs. Two and three-variable response functions are used in the investigation of even and odd-sized point designs. Generated data from designs of sizes 10 and 11 were employed in the investigation. Numerical illustrations were given to ascertain the pattern of convergence on each of the degree polynomial designs. The computations and graphs were conducted in R version 4.1.1 (2021). The results show that cycling patterns differ with respect to the degree of the response function whether it is of even or odd-sized design, or has two or three variables. The result will enable researchers to find appropriate measures to accommodate the challenge posed by cycling.

Universal optimality of circular balanced repeated measurements designs through method of cyclic shifts

The problem of efficiency and universal optimality of RMDs has been widely studied in the literature. Recently, Rajab et al. proposed CBRMDs for equal, two, three, and four different period sizes. In this article, the universal optimality of the proposed designs by Rajab et al., by using the Method of cyclic shifts (Rule I), have been proved under three different models. The information matrices for both direct and residual effects of these designs have also been expressed in the simple form which can be further used for proving conditions of universal optimality. Lastly, the construction of optimal designs through generators has been mentioned.

An algorithm for the simultaneous construction of D-optimal experimental designs

Construction of optimal multiresponse designs when the responses are measured simultaneously for the same settings of the input variables have been studied. However, in some experiments, the response variables can be measured at different time points and also defined in the same experimental space. The response variables are assumed to be uncorrelated. In this case, simultaneous construction of more than one optimal design is required. We present an algorithm for the simultaneous construction of optimal designs for these kind of experiments. The algorithm operates on two parallel platforms. On one, the same support points are selected for the response functions. The starting point of the search, direction vector, and step length are computed. On the other, the choice of initial design and exchange of design points are carried out. Examples are given to illustrate the construction of optimal designs using the algorithm. The algorithm converges rapidly and is more economical than constructing optimal design for one response function at-a-time.

Building some bridges among various experimental designs

Designing their experiments is the significant problem that experimenters face. Maximin distance designs, supersaturated designs, minimum aberration designs, uniform designs, minimum moment designs and orthogonal arrays are arguably the most exceedingly used designs for many real-life experiments. From different perspectives, several criteria have been proposed for constructing these designs for investigating quantitative or qualitative factors. Each of those criteria has its pros and cons and thus an optimal criterion does not exist, which may confuse investigators searching for a suitable criterion for their experiment. Some logical questions are now arising, such as are these designs consistent, can an optimal design via a specific criterion perform well based on another criterion and can an optimal design for screening quantitative factors be optimal for qualitative factors? Through theoretical justifications, this paper tries to answer these interesting questions by building some bridges among these designs. Some conditions under which these designs agree with each other are discussed. These bridges can be used to select a suitable criterion for studying some hard problems effectively, such as detection of (combinatorial/geometrical) non-isomorphism among designs and construction of optimal designs. Benchmarks for reducing the computational complexity are given.

An algorithmic approach to construct D-optimal saturated designs for logistic model

ABSTRACTIn this paper, locally D-optimal saturated designs for a logistic model with one and two continuous input variables have been constructed by modifying the famous Fedorov exchange algorithm. A saturated design not only ensures the minimum number of runs in the design but also simplifies the row exchange computation. The basic idea is to exchange a design point with a point from the design space. The algorithm performs the best row exchange between design points and points form a candidate set representing the design space. Naturally, the resultant designs depend on the candidate set. For gain in precision, intuitively a candidate set with a larger number of points and the low discrepancy is desirable, but it increases the computational cost. Apart from the modification in row exchange computation, we propose implementing the algorithm in two stages. Initially, construct a design with a candidate set of affordable size and then later generate a new candidate set around the points of design searched in the former stage. In order to validate the optimality of constructed designs, we have used the general equivalence theorem. Algorithms for the construction of optimal designs have been implemented by developing suitable codes in R.

Sequential designs for repeated–measures experiments

We discuss optimal sequential designs for analyzing longitudinal data or repeated measurements in the framework of generalized linear mixed models (GLMMs). The construction of optimal designs for GLMMs typically requires intensive computation involving irreducibly high-dimensional integrals with respect to the random effects when calculating the Fisher information. In this article, we propose a simpler method for constructing optimal sequential designs based on some “pseudo” Fisher information. We investigate the performance of the proposed designs through simulation studies. The simulation results indicate that the loss of efficiency from the pseudo-optimal design construction process is minimal as compared to the computationally intensive D-optimal design. Furthermore, the proposed design appears to be as robust as the D-optimal design when possible misspecification to the assumed random effects distribution occurs. It is concluded that the proposed approach is of practical value for designing repeated-measures experiments.

On the construction of optimal designs

Two strategies for specifying additional data to be included with the data of a non-orthogonal design are presented. The additional data increase the magnitude of the information matrix X′X and the orthogonality of the design matrix. Sequentially, the new points are augmented to the original design, such that each new point optimally increases the smallest eigenvalue of X′X. The new runs are created in a predefined spherical region and a rectangular region. Optimum number of additional observations is presented in order to orthogonalize the design matrix X and optimize some functions of the information matrix X′X. Comparisons of the results acquired with the proposed methods versus the most commonly used procedures for data augmentation are carried out. In addition, the advantages of the use of our techniques over the studied methods to solve the augmenting data problems are discussed.

Optimal Crossover Designs for Logistic Regression Models in Pharmacodynamics

Pharmacodynamics (PD) is the study of the biochemical and physiological effects of drugs. The construction of optimal designs for dose-ranging trials with multiple periods is considered in this paper, where the outcome of the trial (the effect of the drug) is considered to be a binary response: the success or failure of a drug to bring about a particular change in the subject after a given amount of time. The carryover effect of each dose from one period to the next is assumed to be proportional to the direct effect. It is shown for a logistic regression model that the efficiency of optimal parallel (single-period) or crossover (two-period) design is substantially greater than a balanced design. The optimal designs are also shown to be robust to misspecification of the value of the parameters. Finally, the parallel and crossover designs are combined to provide the experimenter with greater flexibility.

407 スパッタリングターゲットに関する研究 : スパッタリングターゲットの最適設計手法の構築(GS2-2 加工(2),一般セッション:GS2 加工)

407 スパッタリングターゲットに関する研究 : スパッタリングターゲットの最適設計手法の構築(GS2-2 加工(2),一般セッション:GS2 加工)

Construction of optimal designs using a clustering approach

There are a variety of problems in statistics which require the calculation of one or several optimizing probability distributions or measures. A class of multiplicative algorithms, indexed by functions f ( . ) of derivatives is considered. The performance of the algorithm is first investigated in finding one optimizing distribution, namely a D-optimal design on a continuous compact (design) interval or space. In practice we must discretize these spaces. The optimum design often turns out to be a distribution defined on disjoint clusters of points. These clusters begin to ‘form’ early on in the above iterations. The idea is that, at an appropriate iterate p ( r ) , the single distribution p ( r ) should be replaced by conditional distributions within clusters and a marginal distribution across the clusters. This approach is formulated for a general regression model and, then is explored through several regression models. Considerable improvements in convergence are seen for each of these models.

DS-optimal designs in one way ANOVA

Characterization and construction of optimal designs using the familiar optimality criteria, for example A-, D- and E-optimality are well studied in the literature. However the study of the Distance Optimality (DS-) criterion introduced by Sinha (1970) has very recently drawn attention of researchers. In the present article, we consider the singularly estimable full rank problem of estimating the full set of elementary treatment contrasts using the DS optimality criterion in the set up of a one way ANOVA model. Using a limit argument it turns out that a CRD in which difference between any two allocation numbers is at the most unity is uniquely DS-optimal.

A new design criterion when heteroscedasticity is ignored

This paper examines the construction of optimal designs when one assumes a homoscedastic linear model, but the underlying model is heteroscedastic. A criterion that takes this type of misspecification into account is formulated and an equivalence theorem is given. We also provide explicit optimal designs for single-factor and multi-factor experiments under various heteroscedastic assumptions and discuss the relationship between the D-optimal design sought here and the conventional D-optimal design.

Robust extrapolation designs for biased polynomial models

Abstract In this article we consider the construction of optimal designs for extrapolation in the polynomial regression setup, allowing for imprecision in the specification of the response function. We adopt a minimax approach, which determines an optimal design to minimize the maximum value of the integrated mean-squared prediction error (IMSPE), with the maximum being evaluated over the departures from the model. It turns out that in straight-line and quadratic regression, the minimax extrapolation designs are the same as the minimum integrated variance (MIV) extrapolation designs, which assume that the fitted models are correct. These designs coincide with the D 1 -optimal designs if the extrapolation interval becomes sufficiently large. For cubic polynomial regression, some comparisons of the performance of the proposed designs with uniform and MIV designs are given.

On the design of optimal change-over experiments through multi-objective simulated annealing

The construction of optimal designs for change-over experiments requires consideration of the two component treatment designs: one for the direct treatments and the other for the residual (carry-over) treatments. A multi-objective approach is introduced using simulated annealing, which simultaneously optimises each of the component treatment designs to produce a set of dominant designs in one run of the algorithm. The algorithm is used to demonstrate that a wide variety of change-over designs can be generated quickly on a desk top computer. These are generally better than those previously recorded in the literature.

On the determination and construction of optimal designs for comparing a set of test treatments with a set of controls in the presence of a linear trend

In this paper we consider the problem of constructing optimal designs for experimental situations where c controls are to be compared to t test treatments and the treatments are to be applied to experimental units occurring in a linear array and where there may be an unknown linear trend. Methods are given for determining and constructing optimal designs for such situations.

On optimal allocations for estimating the surface of a random field

In general, the construction of optimal designs is apparently a difficult task for the approximation of a random field indexed by more than one dimension. Besides the rate of convergence of the minimum achievable error hardly anything is known until now. However, if there is an immanent structure present in the random field, then, taking this structure into account, improved estimates can be obtained. For this situation we present adequate designs which show, at least, a nearly optimal performance.

Optimal Row-Column Designs for Multiple Comparisons With a Control: A Complete Catalog

The problem of identification and construction of optimal designs for comparing several test treatments with a control treatment is studied. Two-way heterogeneity in the experimental units is considered under the usual additive model. A general class of designs—named balanced treatment row-column designs—appears to be natural candidates for the A-optimality criterion. Catalogs of A-optimal balanced treatment row-column designs have been prepared and tabulated for a practical range of parameter values. The use of the catalogs is explained with examples.

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.