Construction Of Experimental Designs Under Model Uncertainty

Experimental design is an important branch of statistics. It is not only of great theo-retical importance, but also of great value in practical applications. Classical experimental designs, such as orthogonal designs, regression designs, block designs, Latin square designs and response surface designs, have been studied extensively and have rich theories. These designs are all based on known models with unknown parameters and aim at finding the optimal designs and estimating the paramcters. These designs are called experimental designs under known models. With the rapid developments of science and technology, practical problems arc becoming more and more complicated. and the need for breaking through the model depecndence is getting stronger. Besides. with the development of com-puters. more and more experiments can be carried our by means of computer, which can save experimental expense greatly and quicken the pace of research. There is no random error in the computer experiment. which makes it different from the designs under model uncertainty. However. the two kinds of designs have many in common in designing experi-ments and modeling, thus in this dissertation, both of them are regarded as designs under model uncertainty.This dissertation explores some new subjects of experimental designs under model uncertainty, including the construction of orthogonal Latin hypercube designs (LHDs), orthogonal column designs, nested space-filling designs, sliced space-filling designs and mixed-level supersaturated designs. Besides, a new convenient method of generating gen-eralized minimum aberration (GMA) designs is proposed. Further, we investigate the close relationships among the discrete discrepancy, centered L2-discrepancy (CD2), wrap-around L2-discrepancy (WD2) and generalized wordlength pattern (GWP), and provide some conditions under which a design having one of these minimum discrepancies is equiv-alent to having GMA.In recent years, building surrogate models (or called metamodels) based on computer experiments has been widely used in engineering field. This is primarily because a physics-based model may be represented by a set of complex equations which often have only numerical solutions that are carried out by computer programs and thus cannot sufficiently describe the relationship between the output and input variables. In addition, it is too time-consuming to solve these complex equations. While a metamodel can effectively identify the input-output relationships and save the computation drastically.Since there is no random error in a computer experiment, replication and randomiza-tion are in no need and the uncertainty is only due to a lack of knowledge about the nature of the relationship between the inputs and the outputs. Thus computer experiments need special designs. Most commonly used designs in computer experiments are LHDs, which have one dimensional uniformity and were proposed firstly for computer experiments by McKay, Beckman and Conover (1979). However, the original construction of LHDs by mating factors randomly is susceptible to having potential high correlations among fac-tors. It is desirable to include orthogonal variables in a regression model, so that the estimates of the regression coefficients would be uncorrelated. When fitting the first-order model, the orthogonal LHD ensures the independence of estimates of linear effects. Fur-thermore, it is desirable to have an orthogonal LHD that can estimate the linear effects without being correlated with the estimates of quadratic effects and bilinear interactions, when fitting the first-order model while the second-order effects, i.e. the quadratic effects and bilinear interactions, are present. Thus we seek LHDs with the following properties:(a) the estimates of linear effects of all factors are uncorrelated with each other;(b) the estimates of linear effects of all factors are uncorrelated with the estimates of all quadratic effects and bilinear interactions.There are some existing LHDs with both properties (a) and (b), for example, the L(2c+1,2c) and L(2c+1+1,2c) constructed by Ye (1998); the L(2c+1, c+1+(2c)) and L(2c+1+1, c+ 1+(2c)) for any positive integer c≤11 constructed by Cioppa and Lucas (2007) through extending Ye's procedure; and the LHDs constructed in Georgiou (2009) via (generalized) orthogonal designs, but these designs can accommodate only a few factors. In.addition, the factors in an LHD have as many levels as the run size, which makes it very difficult for an LHD to be orthogonal. Thus, many orthogonal LHDs cannot be constructed by existing methods and even do not exist.To simulate a physical system, one needs to build mathematical models to represent physical behaviors. Models can take different levels of fidelity such as the detailed and the simple models in order to satisfy the experimenter's various demands. Multi-fidelity com-puter experiments are widely used in many engineering and scientific fields. For example, consider the situation involving two-fidelity experiments that are called the low-accuracy experiment and the high-accuracy experiment. The sets of design points for low-accuracy and high-accuracy are denoted by Dl and Dh, respectively. The two-fidelity computer experiments Dl and Dh are desirable to satisfy the following three principles (Qian, Tang and Wu,2009):Economy:the number of points in Dh is smaller than the number of points in Dl(low-accuracy experiment is cheaper than high-accuracy experiment);Nested relationship:Dh is nested within Dl, i.e., Dh(?)Dl(for modeling and calibrating the differences between these two experiments);Space-filling:both Dh and Dl achieve stratification in low dimensions (suppose the inter-esting features of the true model are as likely to be in one part, of the design space as in another).On the other hand, the standard framework for computer experiments assumes that the input factors are quantitative, but some input factors of computer models can be qualita-tive. Qian and Wu (2009) proposed the sliced space-filling design for computer experiments with both qualitative and quantitative factors. It is easy to see that the sliced space-filling design is an experiment with two-fidelity. Design construction for such computer experi-ments is a new issue.Supersaturated design (SSD) is used in the initial stage of an industrial or scientific experiment for screening the active effects, and is useful when there are a large number of factors under investigation while only a very limited number of experimental runs are available. Many optimality criteria have been proposed for design construction and comparison, but the optimal SSDs under these criteria, may include columns with high correlation even fully aliased columns. Thus, it is very necessary to find an easy way to judge whether two columns are fully aliased or not. And it is interesting to construct optimal SSDs with the near-orthogonality between columns being controlled.In addition, a fundamental and practical question for factorial designs is how to choose a good design from a set of candidates. From different viewpoints, various optimality criteria have been proposed. The GMA based on generalized wordlength pattern (GWP) and uniformity measure of discrepancy are two most popular criteria. What are the relationships among them, when are they equivalent to each other? Besides, the generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. A convenient method for the generation of GMA designs is useful both in theory and practice.Based on the above discussions, this dissertation is devoted to the following researches:(1) Constructing LHDs satisfying properties (a) and (b) and can accommodate more fac-tors;(2) By relaxing the condition that the number of levels for each factor must identical to the run size, constructing a class of orthogonal column designs for computer experiments;(3) Constructing designs suitable for the experiments with different fidelities;(4) Constructing sliced space-filling designs for computer experiments with both qualita-tive and quantitative factors;(5) Providing an equivalent condition for two columns to be fully aliased and constructing a scries of optimal SSDs without fully aliased columns and the near-orthogonality between columns being controlled;(6) Proposing a convenient method to generate GMA designs;(7) Exploring the connections between some discrepancies and aberration and showing when they arc equivalent to each other.In the following, let us introduce the contents of each chapter in brief.Chapter 1 summarizes some basic concepts, notation and provides some lemmas that will be used in the following chapters.Chapter 2 is concerned with the construction of LHDs with the properties that the estimates of linear effects of all factors are not only uncorrelated with each other, but also uncorrelated with the estimates of all quadratic effects and bilinear interactions. This chapter completely solves the construction problem for orthogonal LHDs L(2c+1,2c) and L(2c+1+1,2c) with properties (a) and (b). The construction method is convenient and flexible, and the number of factors in any design constructed by the proposed method attains its maximum value among all the corresponding LHDs satisfying properties (a) and (b). Further, we extend the method to construct orthogonal LHDs with more flexible run sizes and properties (a) and (b). At the end of this chapter, we prove that the designs with properties (a) and (b) are optimal under some other meaningful criteria. Chapter 3 introduces a method for constructing a rich class of orthogonal column designs (OCDs) that are suitable for use in computer experiments. For computer experiments, though LHDs have been popular choices, practical experiences have revealed that designs with enough levels are desirable and it is not essential that the run size equals the numbers of factor levels, as in an LHD which is difficult to be orthogonal. Chapter 3 constructs some new OCDs for computer experiments by rotating groups of factors of orthogonal arrays, which supplement the designs for computer experiments in terms of various run sizes and numbers of factor levels and are flexible in accommodating various combinations of factors with different numbers of levels. These OCDs not only have uniformly spaced levels for each factor but also have independent estimates of the linear effects in a first order model. And the estimates of linear effects of all factors are uncorrelated with the estimates of all quadratic effects and bilinear interactions if the corresponding orthogonal arrays have strength equal to or greater than three. Along with a large factor-to-run ratio. these new designs arc economical and suitable for screening factors for physical experimentsChapter 4 presents a method for constructing experiments with two levels of accuracy. Design construction for such computer experiments is a new issue because the existing methods deal almost exclusively with experiments with one level of accuracy, Qian. Tang and Wu (2009) proposed that nested space-filling designs are suitable for conducting such experiments. This chapter proposes a systematic method for constructing nested space-filling designs with two levels of accuracy using the nested difference matrices. These nested space-filling designs can achieve stratification in low dimensions.In Chapter 5, some methods are proposed for constructing nested space-filling designs with multiple levels of accuracy. Chapter 4 has presented some methods for constructing nested space-filling designs with two levels of accuracy or fidelity. But some experiments can often be run at many different levels of sophistication with vastly varying time, for example, these experiments can be a combination of physical experiment, detailed computer experiment and approximate computer experiment. In this chapter, we propose some methods for constructing nested space-filling designs for multiple experiments with different levels of accuracy. These constructions make use of the decomposition of Galois field and the generated designs achieve uniformity in low dimensions. Further, these deigns are sliced space-filling designs, thus can be used in computer experiments with both quantitative factors and qualitative factors.Chapter 6 proposes some methods for constructing E(fNOD)- and/orχ2- optimal mixed-level SSDs without fully aliased columns. SSD has received much interest because of its potential in factor screening experiments. In this chapter, we provide an equivalent condition for two columns to be fully aliased and then propose some methods for constructing E(fNOD)-and/orχ2-optimal large mixed-level SSDs without fully aliased columns from small equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs without fully aliased columns can thus be constructed. Further, we prove that the near-orthogonality between columns of the resulting design, measured by fNOD, is well controlled by that of the source designs, i.e., if the source designs have small values of fNOD, then the resulting design tends to have small values of fNOD. In addition, some newly generated optimal mixed-level:SSDs are tabulated for practical use.In Chapter 7, we provide an algorithmic approach to finding factorial de-signs with GMA. Factorial designs are arguably the most widely used experimental designs in industrial and scientific investigations. Their practical success is due to the efficient use of experimental runs to study many factors simultaneously. One popular cri-terion for selection of fractional factorial designs is GMA. which is based on sequentially minimizing the GWP. The generation of GMA designs is a non-trivial problem due to this sequential optimization nature of the criterion. Based on an analytical expression between the GWP and a general discrepancy provided by Hickernell and Liu (2002), Chapter 6 con-verts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also optimal under the uniformity measure.Chapter 8 explores the connections between discrepancy and aberration in general multi-level factorials. There are many useful criteria defined from different viewpoints for comparing fractional factorial designs. The uniformity measure of discrep-ancy and GMA are two popular and important ones. Among various existing discrepancies, the discrete discrepancy, centered L2-discrepancy(CD2) and wrap-around L2-discrepancy (WD2) have been well justified and widely used. In chapter 8, we express the discrete dis-crepancy, CD2 and WD2 in the second-order polynomials of the indicator functions, and then investigate the relationships between these discrepancies and the GWP. Further, we provide some conditions under which a design having one of these minimum discrepancies is equivalent to having GMA. The close relationships among these criteria further show that the uniformity criteria can be utilized to compare fractional factorial designs and provide an additional rationale for employing uniform designs. All these results show that orthogonality is strongly related to uniformity, and provide some further justifications for the criterion of GMA in terms of uniformity. In addition, the expressions of the discrete discrepancy and WD2 in the quadratic forms of the indicator functions are useful for us to find optimal designs under each of the criteria.