Study On Optimal Foldover Plans Of Fractional Factorials And Related Problems

Experimental design and statistics analysis is one of the most important branches of statistics, it enables an investigator to conduct better experiments, analyze data efficiently, and make the connections between the conclusions from the analysis and the original objectives of the investigation. The frequently-used experimental design includes factorial design, orthogonal design, uniform design, block design, optimal design, response surface design, and so on. With the development of science and technology, a experiment has more and more factors, and each factor's levels are much more than before. In this case, the number of runs in a full factorial de-signs are greatly exceed the researcher's tolerance. For reasons of run size economy, fractional factorial design may be used, which is a part of the full factorial design.In the past two decades, many important theoretical results of fractional facto-rials have been published and widely used in practice. Fractional factorials include regular fractional factorials and nonregular fractional factorials. Regular designs have simple confounding structure, any two effects are either orthogonal or com-pletely aliased with each other. On the other hand, nonregular designs have also been widely used due to their economic and flexible run size.However, one serious consequence of using a fractional factorial design is the aliasing of factorial effects. A standard follow-up strategy based on foldover tech-nique is an important method to break the aliasing in designs. In order to break the aliasing in two-level designs, a standard follow-up strategy is to add a foldover design by reversing the signs-of one or more columns of the initial design. The com-bination of the initial design and its foldover design under a foldover plan is called as a combined design. This idea and technique has been extensively discussed in the literature, and found that foldover design possesses with good structure and statis-tical properties. It is to be noted that existing works on optimal foldover plans are in terms of aberration criterion or clear effects criterion. Fang and Mukerjee (2000) gave an analytic linkage between uniformity and wordlength pattern of a regular two-level designs, firstly showed a link between the two apparently unrelated ar-eas of uniformity and minimum aberration. We can see the minimum aberration criterion and the uniformity criterion are almost equivalent for two-level factorials. Therefore, it is reasonable and feasible to the study of optimal foldover plan bases on uniformity criterion rather than aberration criterion. On the other hand, existing works on foldover design mainly focus on two-level case. The method of reversing the signs of some factors loses its meaning when the original designs involve factors with more than two levels. How to define foldover plan for designs involve factors with more than two levels and study the optimal foldover plan under a suitable criterion? This will be a very important problem.Block design is an important kind of experimental designs. Its basic idea stemmed from agricultural and biological experiment, and now it has been widely applied to science and technology, engineering and many other areas. There are some factors exist in experiment which effect on response but can not be controlled or we don't care, these factors are called as noise factors in treatment. If noise factors are unknown and can not be controlled, we often reduce its impact by randomizing experiment; if noise factors are known and can be controlled, we often reduce its impact by the method of block. Given a blocked design, the study of its optimal foldover plan and related properties, and optimal blocking and foldover plan and related properties of factorials are very important to theory and practice.Doubling is a simple and powerful method to construct two-level fractional fac-torial designs. Through doubling method, we can construct a design with large run size and good properties, such as orthogonal main-effect plans, design of resolution IV or higher, start from a design with small run size. Moreover, Double design has good symmetrical structure and can be regarded as a design constructed by its ini-tial design under a special foldover plan. What are properties of design constructed by its initial design under a general foldover plan? When is it equivalent to Dou-ble design? Besides, the discussions of analytic connections between Double design and its original design in term of various optimal criteria and uniformity of Double design are very meaningful.One of the important tasks in experimental design is to find "good" designs and to analyze experimental data effectively, so that more effects and more possible models related to the effects in experiments can be estimated. In order to define what design is "good", various new optimality criteria have been proposed from different angels or statistical models for design comparison. Based on the viewpoint of orthogonality, Fang, Ma and Mukerjee (2002) proposed B-criterion to measure the orthogonality of symmetrical design. Based on an ANOVA model, Xu and Wu (2001) proposed generalized minimum aberration criterion. Yue (2001) proposed asymptotic Bayes criterion based on a functional ANOVA decomposition model. What relationships among different criteria for optimal factor assignments based on different statistical models?Two-level designs are a kind of most simple and widely used designs. Much attention has been paid to the uniformity of two-level designs in the literature. Discrepancy is applied to measure uniformity in uniform designs and uniform designs favor designs with lower discrepancy. Therefore, the lower bound of discrepancy which measure uniformity is an important benchmark. However, the existing lower bounds of discrepancy can not be attained in many cases, even in two-level case. So how to improve the lower bound of discrepancy is an very important problem, especially for the construction of uniform design.Based on the above discussions, the dissertation is devoted to the following researches:(1) Studying the uniform foldover design for two-level design, generalizing the concept of foldover to design with multi-level factors, discussing the uniform foldover design for asymmetrical designs and obtaining some lower bounds of discrepancy on combined design;(2) Discussing the optimal foldover plans for blocked nonregular two-level de-signs, optimal blocking and foldover plans and properties for nonregular two-level designs;(3) Proposing the concept of generalized Double design and discussing it prop-erties via foldover technique, discussing the analytic connections between Double design and its original design in term of various optimal criteria and uniformity of Double design;(4) Building the connections among asymptotic Bayes criterion, generalized minimum aberration criterion and B-criterion for asymmetrical factorials, and ob-taining the tight lower bound of asymptotic Bayes criterion;(5) Providing a more tighter lower bound of centered L2-discrepancy for regular two-level design and its complementary design. In the following, let us introduce the contents of each chapter in brief.Chapter 1 summarizes some related background of experimental de-sign and the innovation and structure of this dissertation.Chapter 2 briefly introduces some basic concepts, notations and pro-vides some lemmas and results that will be used in the other chapters.Chapter 3 discusses the optimal foldover plans of two-level fractional factorials under uniformity criterion. Because of the close relationship be-tween centered L2-discrepancy and aberration in factorial design (Fang and Muker-jee,2000), the minimum aberration criterion and the uniformity criterion based on centered L2-discrepancy are almost equivalent for two-level factorials. Therefore, it is reasonable to the study of optimal foldover plan based on uniformity criterion. In this chapter, the uniformity of combined design for two-level design under any foldover plan is discussed, some lower bounds of centered L2-discrepancy on com-bined design are obtained. These lower bounds can be used as a benchmark for searching optimal foldover plans.Chapter 4 discusses the optimal foldover plans of mixed-level frac-tional factorials under uniformity criterion. It is indispensable to study the ex-periments with multi-level factors since two-level experimental design is not enough for the practice. In this chapter, the concept of foldover plan is generalized to design with multi-level factors, and the uniformity measured by wrap-around L2-discrepancy of combined design for mixed-level design under any foldover plan is discussed, some lower bounds of wrap-around L2-discrepancy on combined design are obtained. These lower bounds also can be used as a benchmark for searching optimal foldover plans.Chapter 5 discusses some related problems when both block and foldover techniques are applied to nonregular two-level design. Blocking is a commonly used technique to control systematic noises in experiments. For a given blocked regular two-level design, Li and Jacroux (2007) searched by an algorithm the optimal treatment foldover plans under two proposed optimality criteria. Ai, Xu and Wu (2010) considered the optimal plans for regular two-level designs when both blocking and foldover techniques are employed and provided some theoretical insight into the relationships between an initial design and the resulting combined blocked design under a general foldover plan. Using the effective tool of indicator function, this chapter discusses the following two problems:the first one is intended to compliment the work of Li, Lin and Ye (2003) and Li and Jacroux (2007) by con-sidering optimal foldover plans for blocked nonregular two-level designs,the second one is intended to compliment the work of Ai, Xu and Wu (2010) by simultaneously considering optimal blocking and foldover plans for nonregular two-level designs. In addition, the related results of 12,16 and 20 runs nonregular initial designs are tabulated for practical use.Chapter 6 explores the application of foldover technique in Double design and some related problems. In this chapter, we firstly generalize the concept of Double design via foldover technique, propose the concept of generalized Double design and discuss its properties using indicator function, and then discuss the analytic connections between Double design and its original design in term of various optimal criteria, such as E(s2) criterion, minimum moment aberration crite-rion, generalized minimum aberration and minimum projection uniformity criterion. Finally, we also discuss the uniformity of Double design.Chapter 7 gives the connections among asymptotic Bayes criterion, generalized minimum aberration criterion and B-criterion for asymmet-rical factorials, and provides a tight lower bound of asymptotic Bayes criterion. Yue (2001) described a Bayesian model in which the prior for the re-sponse is specified based on a functional ANOVA decomposition, and the asymptotic Bayesian criterion is proposed. Yue and Wu (2004), Yue and Chatterjee (2009) and Yue, Qin and Chatterjee (2011) investigated symmetrical U-type design for non-parametric Bayesian regression with one response or multiresponse under different covariance kernels, respectively. The object of this chapter aims to study the re-lationship of different design criteria via different statistical models and to extend the results in Yue and Wu (2004) and Yue and Chatterjee (2009) from symmetrical factorials to asymmetrical factorials for which we give links between different op-timality measures in the context of fractional factorial experiments under a more general covariance kernel.Chapter 8 provides some more tighter lower bounds of centered L2- discrepancy for regular two-level design and its complementary design. Among various existing discrepancies, centered L2-discrepancy has been well justi-fied and widely used. In this chapter, We express the centered L2-discrepancy as a quadratic form of the coefficients of the indicator function, and obtain some new lower bounds of centered L2-discrepancy for regular two-level design and its comple-mentary design. Numerical results show that our lower bounds are tight and better than the existing results.Chapter 9 summarizes the dissertation and prospects the future work.