Research On Projection Uniformity Of Factorial Designs Based On Mixture Discrepancy

As one of the optimal screening criteria for evaluate the optimality of a design,the uniformity criterion is widely used measure the uniformity of design,such as the centered L2-discrepancy,wrap-around L2-discrepancy and mixture discrepancy,etc.When various discrepancies are used to assess different designs,only the total uniformity of all dimensions is considered,and the uniformity on sub-designs of low dimensions is ignored,which may lead to unreasonable results on low dimensions.Therefore.Hickernell and Liu(2002)proposed the concept of projection discrepancy pattern and minimum projection uniformity criterion,so as to effectively reflect the uniformity of the design in all low-dimensional projections.Recently,there are many important theoretical results in the study of projection uniformity of design.Yi and Zhou(2018)studied the projection uniformity of two-level design,three-level design and two-level combined design based on the mixture discrepancy.These conclusions are only limited to low level design,there is no corresponding research on high level design and asymmetric design.Based on the above discussions,the dissertation is devoted to the following issues:(1)The projection uniformity of mixed two-and three-level designs based on mixture discrepancy is studied.(2)The projection uniformity of q-level design based on mixture discrepancy and its related properties are is discussed.(3)The projection uniformity and the optimal foldover plans of symmetrical combined designs based on mixture discrepancy is explored.(4)The projection uniformity of any symmetrical row augmented design under mixture discrepancy is studied,and the relationship between the uniformity pattern and its lower bound of the row augmented design and the uniformity pattern and its lower bound of initial design and the follow-up design are also given.In the following,let us introduce the contents of each chapter in brief.Chapter 1 summarizes some related background of experimental design and the innovation and structure of this dissertation.Chapter 2 introduces some basic concepts and formulas briefly,and provides some lemmas and results that will be used in other chapters.Chapter 3 mainly studies the projection uniformity of mixed two-and three-level design.The average projection mixture discrepancy of two-and three-level designs is obtained by the method of level permutation.Comparing the average projection mixture discrepancy with orthogonality,the uniformity pattern of two-and three-level design is constructed,and the lower bound of its uniformity pattern is obtained.Chapter 4 mainly discusses the projection uniformity of q-level designs.The uniformity pattern of q-level design is defined,the connections among the uniformity pattern,orthogonality and generalized word-length pattern are established,and a lower bound of the uniformity pattern is obtained.Chapter 5 mainly discusses the projection uniformity of any symmetrical combined designs.The uniformity pattern of symmetrical comb ined design and the optimal foldover plan are defined,the lower bound of its uniform pattern is obtained.Chapter 6 mainly explores the projection uniformity of any symmetrical row augmented designs.The uniformity pattern of symmetrical row augmented design is defined,the lower bound of its uniform pattern is obtained.The relationship between the uniformity pattern and its lower bound of the row augmented design and the uniformity pattern and its lower bound of initial design and the follow-up design are also given.Chapter 7 summarizes the dissertation and prospects in the future work.