Study On Construction Method Of Optimal Fractional Factorial Designs Via Foldover And Code Mapping

Experiment is an important way for people to know,explore and reform the world,it is very vital how to arrange the experiment effectively and to enhance the efficiency for the experiment.Experimental design is one of the most important branches of statistics,which is based on probability theory,mathematical statistics and linear algebra.It does not only scientifically devise the experimental plan,reasonably analyze the experimental results but also obtain sufficiently reliable and useful message by less runs and lower cost.Experimental design has been widely used in industrial and agricultural production,biomedicine,aerospace and so on,and has significantly driven the development of society.When there is more number of factors and levels in one experiment,the cost of arranging the full factorial design may far exceed the tolerance range of people.Hence,it is a better choice to use the fractional factorial design in view of the number of runs and the cost.However,the alias among factorial effects,which leads to some factorial effects can not be effectively distinguish in data analysis,may appear when the fractional factorial design is employed.It is an important method to break the aliasing among factorial effects using foldover technique in follow-up experiments.Many experts and scholars have deeply studied the method on the foldover,and have found that the design which is combined by the initial design and foldover design has good structure and statistical properties,so the technique on foldover has been widely used in the construction of designs.Uniform design is one important kind of fractional factorial designs.Compar-ing to other fractional factorial designs,uniform design provides more choices for experimenter,so it is possible to obtain anticipant results with small runs.Uniform design has been widely used in various fields and has produced notable economic and social benefits since it has been proposed.Uniform design has arranged experiments by uniform design tables,so it is significant to study the construction of uniform design tables.Sudoku design originated from a popular game named Sudoku puzzle has been paid more attention by its special structure.Li,Li and Ou(2014)studied the construction of symmetric uniform design via Sudoku design.In this paper,we extend the result in Li,Li and Ou(2014)to asymmetric uniform design in terms of generalized discrete discrepancy by Sudoku design,foldover and mirror mapping,and obtain a new lower bound of generalized discrete discrepancy for asymmetric designs,which can be used as a benchmark to measaure the uniformity of constructed designs.A common algorithm for constructing asymmetric uniform designs is also proposed,and the related properties of designs by the algorithm are discussed.In order to consider the effectiveness of cost,unreplicated factorial designs are commonly used to identify important or active factorial effects in the early exper-iment stages.But such unreplicated experiments are a big challenge to statistical inference,because no replicates can not estimate the experimental error variance.In this paper,via quarter foldover,we study the construction of the two-level factorials with flexible partially replicated runs,and for an initial design with resolution ?.a(or IV.a),the sufficient and necessary conditions for the constructed design to be a resolution III(IV)or higher design with flexible partially replicated runs are inves-tigated by the tool of indicator function.For the initial designs with 12,16,20 and 24 runs,a catalog of optimal plans to construct some designs with highest resolution and flexible partially replicated runs is also tabulated for practice respectively.Doubling is a simple and effective way to construct two-level fractional factorial designs,which has been used to construct two-level designs with large number of runs and factors and with nice properties,such as orthogonal main-effect plans and designs with high resolution,from two-level designs with small number of runs and factors.In order to construct three-level designs with large number of runs and factors and with nice properties,Zhang(2016)extended the concept of two-level Double design to the three-level Triple design by level permutations,and discussed the properties of Triple designs via indicator function.In this paper,we built the analytic connection between Triple design and its three-level initial design in terms of various design criteria such as E(fNOD)criterion,minimum moment aberration(MMA)criterion,generalized minimum aberration(GMA)criterion and B criterion,and discuss the issue of uniformity for Triple design.Moreover,we extend the concept of three-level Triple design to the four-level Quadruple design,give the structure of Quadruple design,discuss the analytic connections between Quadruple design and its initial design with four-level under various design criteria,and study the uniformity of Quadruple designs.Code theory has been widely used in experimental design,The issue of con-structing optimum designs via code theory has been discussed in many papers.Spe-cially,some scholars have constructed lots of designs with good properties by the code mapping between quaternary codes and binary codes in recent years.We note that the evaluation of designs constructed by the code mapping between quater-nary codes and binary codes is in terms of resolution criterion,aberration criterion and projection criterion,however,it is less in view of uniformity.It is an impor-tant issue how to combine foldover and code mappings between quaternary codes and binary codes.The paper aims to study the following three aspects of code mappings.Firstly,we give the relationship of uniformity between designs with four-level and ones with two-level via two code mappings,which extends and perfects results in Chatterjee et al.(2017).Some improved lower bounds of the wrap-around L2 discrepancy for designs with four-level and two-level are obtained,respectively.Secondly,we propose the concept of four-level Double design,and discuss the con-nections of the uniformity in terms of the wrap-around L2 discrepancy between the four-level Double design and its initial design,the two-level Double design and its initial design,respectively.We also apply the first transformation of two codes map-pings to four-level Double designs and discuss the relationship of uniformity among the four-level design,its corresponding two-level design and their Double designs.Finally,we extend results of four-level Double design to the mixed two-and four-level Double design,and discuss the relationship of uniformity among the mixed two-and four-level design,its corresponding two-level design and their Double designs by combining the first transformation of two codes mappings.