G-optimal Designs Research Articles

Improved G-Optimal Designs for Small Exact Response Surface Scenarios: Fast and Efficient Generation via Particle Swarm Optimization

G-optimal designs are those which minimize the worst-case prediction variance. Thus, such designs are of interest if prediction is a primary component of the post-experiment analysis and decision making. G-optimal designs have not attained widespread use in practical applications, in part, because they are difficult to compute. In this paper, we review the last two decades of algorithm development for generating exact G-optimal designs. To date, Particle Swarm Optimization (PSO) has not been applied to construct exact G-optimal designs for small response surface scenarios commonly encountered in industrial settings. We were able to produce improved G-optimal designs for the second-order model and several sample sizes under experiments with K=1,2,3,4, and 5 design factors using an adaptation of PSO. Thereby, we publish updated knowledge on the best-known exact G-optimal designs. We compare computing cost/time and algorithm efficacy to all previous published results including those generated by the current state-of-the-art (SOA) algorithm, the G(Iλ)-coordinate exchange. PSO is hereby demonstrated to produce better designs than the SOA at commensurate cost. In all, the results of this paper suggest PSO should be adopted by more practitioners as a tool for generating exact optimal designs.

Optimal Design of Experiments on Riemannian Manifolds

ABSTRACT The theory of optimal design of experiments has been traditionally developed on an Euclidean space. In this article, new theoretical results and an algorithm for finding the optimal design of an experiment located on a Riemannian manifold are provided. It is shown that analogously to the results in Euclidean spaces, D-optimal and G-optimal designs are equivalent on manifolds, and we provide a lower bound for the maximum prediction variance of the response evaluated over the manifold. In addition, a converging algorithm that finds the optimal experimental design on manifold data is proposed. Numerical experiments demonstrate the importance of considering the manifold structure in a designed experiment when present, and the superiority of the proposed algorithm. Supplementary materials for this article are available online.

G-optimal designs for hierarchical linear models: an equivalence theorem and a nature-inspired meta-heuristic algorithm

Hierarchical linear models are widely used in many research disciplines and estimation issues for such models are generally well addressed. Design issues are relatively much less discussed for hierarchical linear models but there is an increasing interest as these models grow in popularity. This paper discusses the G-optimality for predicting individual parameters in such models and establishes an equivalence theorem for confirming the G-optimality of an approximate design. Because the criterion is non-differentiable and requires solving multiple nested optimization problems, it is much harder to find and study G-optimal designs analytically. We propose a nature-inspired meta-heuristic algorithm called competitive swarm optimizer (CSO) to generate G-optimal designs for linear mixed models with different means and covariance structures. We further demonstrate that CSO is flexible and generally effective for finding the widely used locally D-optimal designs for nonlinear models with multiple interacting factors and some of the random effects are correlated. Our numerical results for a few examples suggest that G and D-optimal designs may be equivalent and we establish that D and G-optimal designs for hierarchical linear models are equivalent when the models have only a random intercept only. The challenging mathematical question of whether their equivalence applies more generally to other hierarchical models remains elusive.

Optimal Design Generation and Power Evaluation in R: The skpr Package

The R package skpr provides a suite of functions to generate and evaluate experimental designs. Package skpr generates D, I, Alias, A, E, T, and G-optimal designs, and supports custom user-defined optimality criteria, N-level split-plot designs, mixture designs, and design augmentation. Also included are a collection of analytic and Monte Carlo power evaluation functions for normal, non-normal, random effects, and survival models, as well as tools to plot fraction of design space plots and correlation maps. Additionally, skpr includes a flexible framework for the user to perform custom power analyses with external libraries and user-defined functions, as well as a graphical user interface that wraps most of the functionality of the package in a point-and-click web application.

Near G-optimal Tchakaloff designs

We show that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs for polynomial regression. We approximate such designs by a standard multiplicative algorithm, followed by measure concentration via Caratheodory-Tchakaloff compression.

Model-robust G-optimal Designs in the Presence of Block Effects

Model-robust G-optimal Designs in the Presence of Block Effects

Applying the Median and Genetic Algorithm to Construct D- and G-Optimal Robust Designs Against Missing Data

Applying the Median and Genetic Algorithm to Construct D- and G-Optimal Robust Designs Against Missing Data

Fast Computation of Exact G-Optimal Designs Via Iλ-Optimality

ABSTRACTExact G-optimal designs have rarely, if ever, been employed in practical applications. One reason for this is that, due to the computational difficulties involved, no statistical software system currently provides capabilities for constructing them. Two algorithms for exact G-optimal design construction of small designs involving one to three factors have been discussed in the literature: one employing a genetic algorithm and one employing a coordinate-exchange algorithm. However, these algorithms are extremely computer intensive in small experiments and do not scale beyond two or three factors. In this article, we develop a new method for constructing exact G-optimal designs using the integrated variance criterion, Iλ-optimality. We show that with careful selection of the weight function, a difficult exact G-optimal design construction problem can be converted to an equivalent exact Iλ-optimal design problem, which is easily and quickly solved. We illustrate the use of the algorithm for full quadratic models in one to five factors. The MATLAB codes used to implement our algorithm and the exact G-optimal designs produced by the algorithm for each test case are available online as supplementary material.

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.

Optimal and robust designs for trigonometric regression models

This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.

D- and G-optimal experimental designs for the partition coefficient in freeze concentration

A variety of D- and G-optimal experimental designs are developed for continuous freeze concentration experiments when the partition coefficient was determined from the experimental data. The experimental parameters under consideration were the rate of advancement of the solid–liquid interface and the rotation rate of stirring the liquid phase. The data analysis consisted of three model parameters that had to be estimated from the experimental data. The experimental designs developed included simply-supported pseudo-exact D-optimal, pseudo-continuous D-optimal, the corresponding minimax designs, simply-supported G-optimal, and minimax G-optimal designs. The resulting designs show that the minimax G-optimal design can be an efficient design when the model parameters are not well known. When the model parameters are well known, then either the D-optimal or G-optimal designs would be preferred.

Optimal experimental designs in RPLC at variable solvent content and pH based on prediction error surfaces

When pH is used as factor in reversed-phase liquid chromatographic (RPLC) separations, the need for providing quality and informative data with the minimal experimental effort becomes imperative. The most rational way to achieve this is by means of experimental designs. The interest in finding optimal designs involving solvent content and pH in RPLC is considerable, since these factors allow large variations in selectivity when ionisable compounds are involved. Unfortunately, the equations that describe the retention of these compounds with pH are nonlinear. As a consequence, factorial and other designs based on geometrical considerations are not well suited, whereas D-optimal and related designs can only be applied in an iterative fashion. In this work, an extension of G-optimal designs, aimed to enhance the quality of the predictions, is examined for problems involving solvent content and pH. The study was carried out with a set of probe ionisable compounds, for which information on retention behaviour was accurately known. A stepwise strategy was used to obtain a rapid estimation of the best design with a given number of experiments. The objective of the study was to investigate the distribution and number of points in the ideal design for compounds of different acid-base behaviour, and the possibility of finding common designs for groups of compounds. A further goal was to derive design construction rules containing the information requirements, without needing any further mathematical treatment.

Rational Regression Models with Continuous Chebyshev Design

In rational regression models, the G-optimal designs are very difficult to derive in general. Even when an G-optimal design can be found, it has, from the point of view of modern nonparametric regression, certain drawbacks because the optimal design crucially depends on the model. Hence, it can be used only when the model is given in advance. This leads to the problem of finding designs which would be nearly optimal for a broad class of rational regression models. In this article, we will show that the so-called continuous Chebyshev Design is a practical solution to this problem.

Generating and Assessing Exact G-Optimal Designs

We introduce a new algorithm for generating near G-optimal designs for second-order models over cuboidal regions. The algorithm involves the use of Brent's minimization algorithm with coordinate exchange to create designs for 2 to 5 factors. Designs created using this new method either match or exceed the G-efficiency of previously reported designs. We also introduce a new graphical tool, the variance ratio fraction of design space (VRFDS) plot, which is useful for easy comparison of the prediction variance for competing designs over a given region of interest. Using the VRFDS plot to compare G-optimal designs to I-optimal designs shows that the G-optimal designs have higher prediction variance over the vast majority of the design region. This suggests that, for many response surface studies, G-optimal designs may not be the best choice.

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.

Computer-Generated Experimental Designs for Irregular-Shaped Regions

This paper focuses on the construction of computer-generated designs on irregularly-shaped, constrained regions. Overviews of the Fedorov exchange algorithm (FEA) and other exchange algorithms for the construction of D-optimal designs are given. A faster implementation of the FEA is presented, which is referred to as fast-FEA (denoted FFEA). The FFEA was applied to construct D-optimal designs for several published examples with constrained experimental regions. Designs resulting from the FFEA are more D-efficient than published designs, and provide benchmarks for future comparisons of design construction algorithms. The construction of G-optimal designs for constrained regions is also discussed and illustrated with a published example.

Properties of optimal polynomial designs under smoothing spline models

This paper investigates the properties of G-optimal polynomial designs under smoothing spline models. A general formula is derived for the variance function of G-optimal polynomial designs and other designs with minimal support size. Moreover, it is shown by a sequence of propositions that G-optimal polynomial designs remain G-optimal under smoothing spline models for a subclass of models. The results are highlighted in the special cases of cubic and quintic smoothing splines.

Restricted optimal designs for linear and quadratic polynomial regressions

The aim of restricted D- and G-optimal designs introduced by Stigler is to deal with the linear regression model with the possible occurrence of a quadratic term. In this paper, the idea has been extended to A-, and Q-optirnalities as well as to general polynomial regression models. Some interesting results have been found in the case of linear regression with the possible occurrence of quadratic and cubic terms. In some situations, the so-called Ck-restricted optimal designs for this regression model can be solved explicitly.

On the equivalence of D and G-optimal designs in heteroscedastic models

Conditions are derived for the Kiefer-Wolfowitz's theorem (KWT) to apply to linear models with heteroscedastic errors. It is shown that both D- and G-optimal designs remain equivalent only under very stringent conditions. Examples are constructed and their efficiencies are compared when they are not equivalent. For the simple linear regression model with a symmetric efficiency function, a relationship between the efficiencies of the optimal designs and the support points of the D-optimal design is noted.

Comparing robust properties of A, D, E and G-optimal designs

We examine the A, D, E and G-efficiencies of using the optimal design for the polynomial regression model of degree k when the hypothesized model is of degree j and 1⩽ j⩽ k⩽8. The robustness properties of each of these optimal designs with respect to the other optimality criteria are also investigated. Relationships among these efficiencies are noted and practical implications of the results are discussed. In particular, our numerical results show E-optimal designs possess several properties not shared by the A, D and G-optimal designs.