T-optimal Designs Research Articles

Computing T-optimal designs via nested semi-infinite programming and twofold adaptive discretization

Modelling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the (design) points from a generally continuous design space at which the models deviate most from each other under the condition that they are best fitted to those points. Thus, the T-criterion represents a bi-level optimization problem, which can be transferred into a semi-infinite one but whose solution is very unstable or time consuming for non-linear models and non-convex lower- and upper-level problems. If one considers only a finite number of possible design points, a numerically well tractable linear semi-infinite optimization problem arises. Since this is only an approximation of the original model discrimination problem, we propose an algorithm which alternately and adaptively refines discretizations of the parameter as well as of the design space and, thus, solves a sequence of linear semi-infinite programs. We prove convergence of our method and its subroutine and show on the basis of discrimination tasks from process engineering that our approach is stable and can outperform the known methods.

The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs

In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.

T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples.

T-optimal designs for discrimination between rational and polynomial models

T-optimal designs for discrimination between rational and polynomial models

T-optimal designs for discrimination between rational and polynomial models

This paper considers the problem of the analytical construction of experimental designs optimal with respect to the popular T-optimality criterion proposed by A.C. Atkinson and V.V. Fedorov in 1975 for discrimination between the simplest rational and polynomial regression models. It is shown how the classical results from approximation theory can be used to derive explicit formulas describing the behavior of support points and weights of T-optimal designs for different fixed parameter values. An applied discrimination problem for rational and polynomial regression models is considered as an example. For this models the numerical construction of experimental designs optimal with respect to robust analogues of T-criterion is also briefly discussed.

A Semi-Infinite Programming based algorithm for determining T-optimum designs for model discrimination.

T-optimum designs for model discrimination are notoriously difficult to find because of the computational difficulty involved in solving an optimization problem that involves two layers of optimization. Only a handful of analytical T-optimal designs are available for the simplest problems; the rest in the literature are found using specialized numerical procedures for a specific problem. We propose a potentially more systematic and general way for finding T-optimal designs using a Semi-Infinite Programming (SIP) approach. The strategy requires that we first reformulate the original minimax or maximin optimization problem into an equivalent semi-infinite program and solve it using an exchange-based method where lower and upper bounds produced by solving the outer and the inner programs, are iterated to convergence. A global Nonlinear Programming (NLP) solver is used to handle the subproblems, thus finding the optimal design and the least favorable parametric configuration that minimizes the residual sum of squares from the alternative or test models. We also use a nonlinear program to check the global optimality of the SIP-generated design and automate the construction of globally optimal designs. The algorithm is successfully used to produce results that coincide with several T-optimal designs reported in the literature for various types of model discrimination problems with normally distributed errors. However, our method is more general, merely requiring that the parameters of the model be estimated by a numerical optimization.

Robust $T$-optimal discriminating designs

This paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the T-optimality criterion introduced by Atkinson and Fedorov [Biometrika 62 (1975) 57-70]. T-optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification. As a solution to this problem we propose a Bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the T-optimality criterion. It is shown that the corresponding Bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria. For the problem of discriminating between two polynomial regression models which differ in the degree by two the robust T-optimal discriminating designs can be found explicitly. The results are illustrated in several examples.

Optimal Designs for Two Nested Pharmacokinetic Models with Correlated Observations

Two nested pharmacokinetic models are considered in this article. Several observations are taken on the same subject so they are correlated. The covariance function assumed is an exponential covariance function. Optimal exact designs are computed with different criteria, both for discriminating between models and estimating parameters. Compound criteria to estimate the parameters and nonlinear functions of the parameters are used. An iterative algorithm based on T-optimality and an algorithm from Brimkulov et al. (1986) are combined in order to compute T-optimal designs with correlated observations. Finally, compound designs to discriminate between the models and estimate the nonlinear functions are considered. A test power study is performed to adjust the compound parameter.

T-optimal designs for discrimination between two polynomial models

This paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree $n-2$ and $n$. In a fundamental paper, Atkinson and Fedorov [Biometrika 62 (1975a) 57--70] proposed the $T$-optimality criterion for this purpose. Recently, Atkinson [MODA 9, Advances in Model-Oriented Design and Analysis (2010) 9--16] determined $T$-optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide half of the circle into equal parts if the coefficient of $x^{n-1}$ in the polynomial of larger degree vanishes. In the present paper we give a strong justification of the conjecture and determine all $T$-optimal designs explicitly for any degree $n\in\mathbb{N}$. In particular, we show that there exists a one-dimensional class of $T$-optimal designs. Moreover, we also present a generalization to the case when the ratio between the coefficients of $x^{n-1}$ and $x^n$ is smaller than a certain critical value. Because of the complexity of the optimization problem, $T$-optimal designs have only been determined numerically so far, and this paper provides the first explicit solution of the $T$-optimal design problem since its introduction by Atkinson and Fedorov [Biometrika 62 (1975a) 57--70]. Finally, for the remaining cases (where the ratio of coefficients is larger than the critical value), we propose a numerical procedure to calculate the $T$-optimal designs. The results are also illustrated in an example.

Designing experiments to differentiate between adsorption isotherms using T-optimal designs

T-optimal experimental designs are developed to distinguish between the 3-parameter Guggenheim–Anderson–deBoer (GAB) and 2-parameter Brunauer–Emmett–Teller (BET) adsorption isotherms. The results show that the designs are not dependent on one of the GAB model parameters (the monolayer capacity, which is a linear scaling factor), but are dependent on the other two, which determine the curvature and shape of the isotherms. Further analysis of the designs show that for the range of model parameters considered, misspecification of the model parameters can reduce the efficiency of the experimental design to 30% or less: proper selection of the model parameters at which the design is developed within the model-parameter space can increase the maximum–minimum design efficiency to nearly 55%. The experimental replication inherent in T-optimal designs is then used in combination with a statistical lack-of-fit analysis to illustrate the role of experimental variability on the ability to distinguish between the two isotherm models.

Optimal discrimination designs

We consider the problem of constructing optimal designs for model discrimination between competing regression models. Various new properties of optimal designs with respect to the popular T-optimality criterion are derived, which in many circumstances allow an explicit determination of T-optimal designs. It is also demonstrated, that in nested linear models the number of support points of T-optimal designs is usually too small to estimate all parameters in the extended model. In many cases T-optimal designs are usually not unique, and in this situation we give a characterization of all T-optimal designs. Finally, T-optimal designs are compared with optimal discriminating designs with respect to alternative criteria by means of a small simulation study.