Optimal designs for statistical analysis with Zernike polynomials

The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and Φ _p -optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849–879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's Φ _p -criteria (p>?∞) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.     

Optimal designs for calibration of orientations

This paper concerns designed experiments involving observations of orientations following the models of Prentice (1989) and Rivest &Chang (2006). The authors state minimal conditions on the designs for consistent least squares estimation of the matrix parameters in these models. The conditions are expressed in terms of the axes and rotation angles of the design orientations. The authors show that designs satisfying U₁ + … + U_n = 0 are optimal in the sense of minimizing the estimation error average angular distance. The authors give constructions of optimal n‐point designs when n ≥ 4 and they compare the performance of several designs through approximations and simulation.     

Optimal designs for treatment comparisons represented by graphs

Consider an experiment for comparing a set of treatments: in each trial, one treatment is chosen and its effect determines the mean response of the trial. We examine the optimal approximate designs for the estimation of a system of treatment contrasts under this model. These designs can be used to provide optimal treatment proportions in more general models with nuisance effects. For any system of pairwise treatment comparisons, we propose to represent such a system by a graph. Then, we represent the designs by the inverses of the vertex weights in the corresponding graph and we show that the values of the eigenvalue-based optimality criteria can be expressed using the Laplacians of the vertex-weighted graphs. We provide a graph theoretic interpretation of D-, A- and E-optimality for estimating sets of pairwise comparisons. We apply the obtained graph representation to provide optimality results for these criteria as well as for ’symmetric’ systems of treatment contrasts.     

Optimal block designs comparing treatments with a control when the errors are correlated

The efficient design of experiments for comparing a control with v new treatments when the data are dependent is investigated. We concentrate on generalized least-squares estimation for a known covariance structure. We consider block sizes k equal to 3 or 4 and approximate designs. This method may lead to exact optimal designs for some v, b, k, but usually will only indicate the structure of an efficient design for any particular v, b, k, and yield an efficiency bound, usually unattainable. The bound and the structure can then be used to investigate efficient finite designs.     

Optimal regular graph designs

A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). To explore the landscape of designs with a large number of blocks, we introduce extensions of regular graph designs. These are constructed by adding the blocks of a balanced incomplete block design repeatedly to the original design. We present the results of an exact computer search for the best regular graph designs and the best extended regular graph designs with up to 20 treatments v, block size \(k \le 10\) and replication r \(\le 10\) and \(r(k-1)-(v-1)\lfloor r(k-1)/(v-1)\rfloor \le 9\).     

On D-optimal robust designs for lifetime improvement experiments

Locating the optimal operating conditions of the process parameters is critical in a lifetime improvement experiment. For log-normal lifetime distribution with compound error structure (i.e., symmetry, inter-class and intra-class correlation error structures), we have developed methods for construction of D-optimal robust first order designs. It is shown that D-optimal robust first order designs are always robust first order rotatable but the converse is not always true.     

Optimal designs for tumor regrowth models

Most growth curves can only be used to model the tumor growth under no intervention. To model the growth curves for treated tumor, both the growth delay due to the treatment and the regrowth of the tumor after the treatment need to be taken into account. In this paper, we consider two tumor regrowth models and determine the locally D- and c-optimal designs for these models. We then show that the locally D- and c-optimal designs are minimally supported. We also consider two equally spaced designs as alternative designs and evaluate their efficiencies.     

Optimal two-point designs for the michaelis-menten model with heteroscedastic errors

We construct D-optimal designs for the Michaelis-Menten model when the variance of the response depends on the independent variable. However, this dependence is only partially known. A Bayesian approacn is used to find an optimal design by incorporating the prior lnformation about the variance structure. We demonstrate the method for a class of error variance structures and present efficiencies of these optimal designs under prior mis-specifications. In particular, we show that an erroneous assumption on the variance structure for the Michaelis-Menten model can have serious consequences.     

Optimal U-type design for Bayesian nonparametric multiresponse prediction

This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.     

Optimal balanced block designs for correlated observations

The construction of universally optimal designs, if such exist, is difficult to obtain, especially when there are some nuisance effects or correlated errors. The hub correlation is a special correlation structure with applications to experiments in genetics, networks and other areas in industry and agriculture. There may be restrictions on the correlation values of the hub structure depending on the experiment. Optimality of block designs under hub correlation has been studied for the case of a constant correlation value. In this article, we consider the hub structure when one of the correlation values is different from the others, and the universally optimal block designs, binary or non-binary, are theoretically obtained. Also, we introduce an algorithm to construct the optimal designs. The Canadian Journal of Statistics 48: 596–604; 2020 © 2020 Statistical Society of Canada     

Optimal designs for approximating the path of a stochastic process

We consider a centered stochastic process {X(t):t ∈ T} with known and continuous covariance function. On the basis of observations X(t₁), …, X(t_n) we approximate the whole path by orthogonal projection and measure the performance of the chosen design d = (t₁, …, t_n)′ by the corresponding mean squared L²-distance. For covariance functions on T² = [0, 1]², which satisfy a generalized Sacks-Ylvisaker regularity condition of order zero, we construct asymptotically optimal sequences of designs. Moreover, we characterize the achievement of a lower error bound, given by Micchelli and Wahba (1981), and study the question of whether this bound can be attained.     

Optimal designs for second-degree Kronecker model mixture experiments

The paper investigates optimal designs in the second-degree Kronecker model for mixture experiments. Three groups of novel results are presented: (i) characterization of feasible weighted centroid designs for a maximum parameter system, (ii) derivations of D-, A-, and E-optimal weighted centroid designs, and (iii) numerically φ_p-optimal weighted centroid designs. Results on a quadratic subspace of invariant symmetric matrices containing the information matrices involved in the design problem serve as a main tool throughout the analysis.     

Rotation designs for experiments in high-bias situations

Many experiments in the physical and engineering sciences study complex processes in which bias due to model inadequacy dominates random error. A noteworthy example of this situation is the use of computer experiments, in which scientists simulate the phenomenon being studied by a computer code. Computer experiments are deterministic: replicate observations from running the code with the same inputs will be identical. Such high-bias settings demand different techniques for design and prediction. This paper will focus on the experimental design problem introducing a new class of designs called rotation designs. Rotation designs are found by taking an orthogonal starting design D and rotating it to obtain a new design matrix D_R=DR, where R is any orthonormal matrix. The new design is still orthogonal for a first-order model. In this paper, we study some of the properties of rotation designs and we present a method to generate rotation designs that have some appealing symmetry properties.     

Screening designs for model discrimination

We introduce new criteria for model discrimination and use these and existing criteria to evaluate standard orthogonal designs. We show that the capability of orthogonal designs for model discrimination is surprisingly varied. In fact, for specified sample sizes, number of factors, and model spaces, many orthogonal designs are not model discriminating by the definition given in this paper, while others in the same class of orthogonal designs are. We also use these criteria to construct optimal two-level model-discriminating designs for screening experiments. The efficacy of these designs is studied, both in terms of estimation efficiency and discrimination success. Simulation studies indicate that the constructed designs result in substantively higher likelihoods of identifying the correct model.     

Optimal designs for covariates’ models

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Optimal designs for stated choice experiments that incorporate ties

In 1970 Davidson generalised the Bradley–Terry model to allow respondents to say that the two options presented in a choice task were equally attractive. In this paper we extend this idea to the MNL model with m options in each choice set and we show that the optimal designs for the MNL model are also optimal in this setting.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.     

Robust prediction and extrapolation designs for censored data

In this paper we present the construction of robust designs for a possibly misspecified generalized linear regression model when the data are censored. The minimax designs and unbiased designs are found for maximum likelihood estimation in the context of both prediction and extrapolation problems. This paper extends preceding work of robust designs for complete data by incorporating censoring and maximum likelihood estimation. It also broadens former work of robust designs for censored data from others by considering both nonlinearity and much more arbitrary uncertainty in the fitted regression response and by dropping all restrictions on the structure of the regressors. Solutions are derived by a nonsmooth optimization technique analytically and given in full generality. A typical example in accelerated life testing is also demonstrated. We also investigate implementation schemes which are utilized to approximate a robust design having a density. Some exact designs are obtained using an optimal implementation scheme.     

Optimal block sequences for blocked fractional factorial split-plot designs

It is known that for blocked 2^n-k$2^{n-k}$ designs a judicious sequencing of blocks may allow one to obtain early and insightful results regarding influential parameters in the experiment. Such findings may justify the early termination of the experiment thereby producing cost and time savings. This paper introduces an approach for selecting the optimal sequence of blocks for regular two-level blocked fractional factorial split-plot screening experiments. An optimality criterion is developed so as to give priority to the early estimation of low-order factorial effects. This criterion is then applied to the minimum aberration blocked fractional factorial split-plot designs tabled in McLeod and Brewster [2004. The design of blocked fractional factorial split-plot experiments. Technometrics 46, 135–146]. We provide a catalog of optimal block sequences for 16 and 32-run minimum aberration blocked fractional factorial split-plot designs run in either 4 or 8 blocks.