Construction And Application Of Uniform Augmented Designs Based On Lee Discrepancy

How to reasonably design the scale of an experiment is always taken into con-sideration in design of experiment.The participant has a further understanding of the current experiment by the process of experimentation and modeling.However,the expected purpose of the experiments could not achieve through one experiment usually.Therefore,the follow-up experiment is often arranged to get more effective information after the initial experiment,and then the whole process of experiment is continuously improved.Hence,follow-up strategy is widely used in the practical application such as science and engineering,since it is an indispensable strategy.Recently,one of the popular follow-up strategies is proposed.This strategy constructs the row augmented design by adding any runs of follow-up experiment based on the initial experiment.Additionally,the column augmented design can be constructed by adding any number of factors based on the row augmented design,and the level of added factors can be either two-level or three-level.The uniformity of above two kinds of augmented designs is measured by the Lee discrepancy.Lee discrepancy is a method to measure the uniformity of a design,which has good properties.The lower bound of Lee discrepancy is a very important issue in both mathematical and statistical viewpoints.Therefore,the issues about the lower bound of Lee discrepancy have attracted the attention of many scholars in recent years.If there is a uniform design whose squared Lee discrepancy value can reach a lower bound then this lower bound is called a strict lower bound.In return,a tight lower bound can be used to verify the uniformity of a given design.If the squared Lee discrepancy value of a given design is closer to the lower bound of Lee discrepancy,the uniformity of the design is better.Based on the above discussions,the dissertation is devoted to the following researches:(1)The uniformity of the symmetrical row augmented designs is discussed.The quantitative relationships of lower bounds of Lee discrepancy between the row augmented designs and its initial designs or the follow-up designs are built.(2)The uniformity of the asymmetrical row augmented designs is discussed.The quantitative relationships of lower bounds of Lee discrepancy between the row augmented designs and its initial designs or the follow-up designs are built.(3)As for the asymmetrical designs,the construction theory about the col-umn augmented design is discussed based on the row augmented design.And the uniformity of the column augmented design is measured by the Lee discrepancy.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,formulas that will be used in the other chapters.Chapter 3 mainly studies the construction theory about the sym-metrical uniform row augmented designs and the lower bounds of the Lee discrepancy.The symmetrical two-level,three-level,four-level and five-level row augmented designs are discussed,respectively.The uniformity of the row aug-mented designs is measured by the Lee discrepancy.The quantitative relationships of the lower bounds of Lee discrepancy between the row augmented designs and its initial designs or the follow-up designs are built in this chapter.Chapter 4 mainly studies the construction theory about the asym-metrical uniform augmented design and the lower bounds of the Lee dis-crepancy.The construction theory about the uniform row augmented design and the uniform column augmented design is discussed for the mixed two-and three-level design.And the adding factors of the uniform column augmented design can either be two-level or three-level.The uniformity of the augmented design is mea-sured by the Lee discrepancy.The quantitative relationships of the lower bounds of Lee discrepancy between the row augmented designs and the initial designs or the follow-up designs are built in this chapter.Chapter 5 gives the construction algorithm and some examples of the application by the theoretical results in Chapters 3 and 4.Chapter 6 summarizes the dissertation and prospects in the future work.