Projection Uniformity Of Nearly Balanced Designs

In the design of experiments,with the increase of the number of factors and levels,how to reasonably design the scale of an experiment becomes an important research direction of experiment designs.Real-world simulation studies is limited by time and cost,some experiments are non-repeatable and non-randomization,such as clinical trials of chemical drugs and biological products,so it is necessary to construct efficient nearly balanced designs.In order to obtain efficient nearly balanced designs,the projection uniformity of the designs needs to be considered.The uniform projection criterion has good property in measuring the projection uniformity of the designs.At the same time,uniform projection designs const ructed by the uniform projection criterion have good space-filling properties in orthogonality,uniformity and projection uniformity,the unifom projection criterion can be used to construct efficient designs.For agiven design,the uniform projection criterion is defined on the projection uniformity measure at the same time,the lower bound of ?(d)can be used to check the projection uniformity of a given design.If the value of ?(d)for a given design is closer to the lower bound for ?(d),then the projection uniformity of the design is better.Based on the above discussions,the dissertation is devoted to the following researches:(1)The projection uniformity measure ?(d)and the uniform projection criterion of symmetric nearly balanced designs are defined,and the specificity calculation formula of ?(d)and the lower bound of ?(d)are obtained,the projection uniformity of the symmetric nearly balanced designs is discussed.The quantitative relationship of ?(d)between the symmetric nearly balanced designs and the symmetric balanced designs is established.(2)The projection uniformity measure ?(d)and the uniform projection crite-rion of asymmetric nearly balanced designs ard defined,the specificity calculation formula of ?(d)and the lower bound of ?(d)are obtained,the lower bound of ?(d)is used to measure the projection uniformity of the asymmetric nearly balanced designs.The quantitative relationship of ?(d)between the symmetrical nearly bal-anced designs and the symmetrical balanced designs is established,at the same time,the relationship of ?(d)between the symmetric nearly balanced designs and the asymmetric nearly balanced designs is established.(3)The construction theory and examples of symmetric and asymmetric nearly balanced designs with optimal projection uniformity are built.In the following,this thesis 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,formulas that will be used in the other chapters.Chapter 3 mainly study the projection uniformity of the two-level nearly balanced designs.The projection uniformity measure ?(d)is defined based on the wrap-around L2-discrepancy,the uniform projection criterion of two-level nearly balanced designs is defined,and the algorithm of ?(d)and the lower bound of ?(d)are obtained.Finally,an example is given to measure the projection uniformity of such designs by using the lower bound.Chapter 4 mainly study the projection uniformity of the three-level nearly balanced designs.The projection uniformity measure ?(d)is defined based on the wrap-around L2-discrepancy,the uniform projection criterion of three-level nearly balanced designs is defined,and the algorithm of ?(d)and the lower bound of ?(d)are obtained.Finally,some examples are given to measure the projection uniformity of such designs by using the lower bound.Chapter 5 study the projection uniformity of mixed two-and three-level nearly balanced designs.The projection uniformity measure ?(d)is defined based on the wrap-around L2-discrepancy,the uniform projection criterion of mixed two-and three-level nearly balanced designs is defined,and the algorithm of ?(d)and the lower bound of ?(d)are obtained.Finally,an example is given to measure the projection uniformity of such designs by using the lower bound.Chapter 6 gives the construction algorithm and some examples of the application by the theoretical results in Chapters 3,Chapters 4 and 5.In this chapter,the lower bound of ?(d)of the nearly balanced designs is used to measure the projection uniformity of the designs.an nearly balanced designs con-struction method is proposed.The nearly balanced designs of the optimal project ion uniformity are constructed.Chapter 7 summarizes the dissertation and prospects in the future work.