| Experimental design and statistical analysis has rapidly developed to be one of the most important branches of statistics, ever since R. A. Fisher found the modern discipline, and it has been successfully applied in many fields of scientific investiga-tion, including agriculture, medicine, and behavioral research as well as chemical, manufacturing and high-tech industries. It enables the investigator to conduct bet-ter experiments, analyze data efficiently, and make the connections between the conclusions from the analysis and the original objectives of the investigation. With the development of science and technology, an experiment has more and more fac-tors, and each factor’s levels are much more than before. In this case, the number of runs in a full factorial designs greatly exceeds the researcher’s tolerance. For reasons of run size economy, fractional factorial design may be used, which is part of a full factorial design. So now it is the core problem for fractional factorial design that how to select the right test points from a full factorial design by an economic and effective way, namely, how to select an "optimal" design from candidate designs with the same size.There are many different kinds of design screening criteria to assess and com-pare the goodness of factorial designs based on different considerations. For exam-ple, criteria of maximum resolution and (generalized) minimum aberration consider confounding situation between treatment effects, minimum moment aberration cri-terion studies similarities between different runs, orthogonality criterion (also known as B-criterion) considers level combinations’ balance of different factors, uniformity criterion (also known as discrepancy criterion) considers the estimation of the over-all mean from a multivariate quadrature perspective. Among these criteria, the uniformity criterion is suitable for any kind of designs, and has advantages of being model robust, run size flexible et. al. So it has been generally recognized and largely studied in recent years. In fact, discrepancy criterion measures uniformity of design points scattered over the the experimental domain from a geometric angle. It seems completely irrelevant with other criteria, but more and more literatures demonstrate that they are closely related.When we use discrepancy criterion to assess different designs, we usually only consider the total uniformity of all dimensions, but ignore uniformity on sub-designs of low dimensions, which may lead to unreasonable results. With the same motiva-tion for maximum resolution criterion and minimum aberration criterion, we make up a uniformity pattern vector like the word-length pattern by projection unifor-mity namely projection discrepancy on each dimension, which is an important tool to measure the abilities of projected designs onto different dimensions. According to the principle of low-dimensional projection uniformity is more important, natu-rally we prefer designs which sequentially minimize the uniformity pattern, this is the criterion of minimum projection uniformity. And the optimal design under this criterion is called a minimum projection uniform design. Hickernell and Liu (2002) proposed the concept of projection discrepancy pattern based on the unified form of discrepancy. They proved that the projection discrepancy pattern was equivalent to the generalized word-length pattern when a special kernel was used to measure the uniformity, which showed that the generalized word-length pattern was just a special case of the projection discrepancy pattern, so, the research of projection discrepancy pattern will be more valuable in theory and practice.Recently, much work has been done on the topic of projection uniformity, but most aims at2-level designs. For example, Fang and Qin (2005) gave the specific calculation formula of projection discrepancy for designs in D(n;2s) based on the centered L2-discrepancy, and the concept of uniformity pattern and related crite-ria of minimum projection uniformity (MPU) and maximum uniformity resolution were the first time proposed explicitly to assess2-level factorials in it. But for symmetrical designs with q (q≥2) or asymmetrical designs, few work has been done about the projection uniformity and its application. Centering on the theme of uniformity pattern and related criteria, this dissertation tried to go on with the work of Hickernell and Liu (2002) and Fang and Qin (2005), improve the theory of uniformity pattern and MPU criterion. We defined the concept of projection discrepancy and uniformity pattern for different design classes based on different discrepancy according to their own characteristics, for D(n; qs)(q≥2) based on discrete discrepancy D(P;a,6), for D(n;2s1×3s2) based on Lee discrepancy and for D(n; q1×…×qs) based on discrete discrepancy D(P;γ). And respectively we deduced the the calculation formula of uniformity pattern, and proposed the criteria of maximum uniformity resolution and minimum projection uniformity (MPU for short). For any design class above mentioned, we discover the MPU criterion is closely related with some traditional design screening criteria such as GMA, MMA and orthogonality criteria which may be verified by analytical relationship between uniformity pattern and generalized word-length pattern, B-vector. These close rela-tionships provide statistical justification for superiority of the MPU criterion on the one hand, and on the other hand lend a further justification for GMA, MMA and orthogonality criteria from a projection uniformity interpretation, enrich the theory research of experiment design. The definition of projection discrepancy is different from that in Hickernell and Liu (2002) and Fang and Qin (2005), so the study for D(n;q1×…×qs) is totaly different from Hickernell and Liu (2002) both in contents and methods.We know that under the discrepancy criterion a good lower bound plays a key role in searching and constructing uniform designs. Similarly, it is also significant to get a proper lower bound for projection discrepancy on each dimension, which plays as a benchmark for searching MPU designs. This dissertation aims at uniformity pattern on design classes of D(n; qs)(q≥2),D(n;2s1×3s2) and D(n; q1×…qs), finds out precise and strict lower bounds respectively, which would be verified to be tight by some examples.Based on design efficiency measured by maximum estimation capacity criterion under model uncertainty, we studied the relationship between MPU criterion and design efficiency criterion. We find that they are analytically connected for orthog-onal array of strength two, which supplies a further justification for MPU criterion from a perspective of model robustness.Complementary design theory is an important technic in experiment design. It will be doubly beneficial to describe characteristics of original design through its complementary design especially when the complementary design is much smaller. Song and Qin (2010) discussed the application of MPU criterion in2-level comple-mentary deigns, in this dissertation we extend their results, discuss application of MPU criterion in q-level complementary deigns. We try to characterize projection uniformity for q-level designs in terms of their complementary designs, and get their analytical relationship. We also propose a much more convenient and simpler rule of MPU and illustrate it by some numerical examples. In the following, let us introduce the contents in each chapter of this dissertation in brief.Chapter1summarizes some related background of experimental de-sign and the innovation and structure of this dissertation.Chapter2briefly introduces some basic concepts, notations and pro-vides some lemmas and results that will be used in the other chapters.Chapter3,4and5is the body of this dissertation with the same structure. We defined projection discrepancy and uniformity pattern for different design classes based on different discrepancy according to their own characteristics. Based on the uniformity pattern, we proposed criteria of maximum uniformity resolution and minimum projection uni-formity, discussed its statistical justification and some tight lower bounds for uniformity pattern. Different design classes are different chapters. Chapter3discusses uniformity pattern and related criteria for D(n; qs)(q≥2) based on dis-crete discrepancy D(P; a, b), Chapter4for D(n;2s1×3s2) based on Lee discrepancy and Chapter5for D(n; q1×…×qs) based on discrete discrepancy D(P;γ). In each chapter, we gave a calculation formula for the corresponding uniformity pattern and analytical relationship between it and generalized word-length pattern, B-vector. A strict lower bound also is given in each chapter. In chapter5, we additionally discussed some properties of discrete discrepancy D(P;γ). Finally, every chapter was ended by two or three illustrative examples.Chapter6discusses relationship between criteria of MPU and design efficiency. Qin, Zou and Zhang (2011) discussed the relationship for two-level factorials, which showed for orthogonal array of strength two, the two criteria of MPU and design efficiency were almost equivalent. Chapter6extends their results, discussed the relationship of the two criteria for q-level factorials, which verified the statistical rationality of MPU criterion from a perspective of model robustness.Chapter7discusses the application of MPU criterion in complemen-tary designs. We extend results of Song and Qin (2010), try to express projection uniformity of the original design through its complementary design. This comple-mentary technic is quite efficient when the size of complementary design is relatively small, which is verified by some illustrative examples.Chapter9summarizes the dissertation and prospects the future work. |
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