Construction Of Some Space-filling Designs And Screening Designs

Experiments are important tools for people to understand and investigate the truth of nature,which have important theoretical significance and application value in industry,agriculture,engineering,science and many other fields.Experiments are mainly classified as physical experiments and computer experiments.A physical experiment is traditionally implemented in an agricultural field,a factory or a laboratory where the experimenter physically carries out the experiment and makes field observations,while a computer experiment is implemented by a complex computer code.Experimental design is one of the most important steps in the experiment,and some attractive properties in the statistical analysis can be achieved by reasonably controlling the value of variables.This dissertation concentrates on some new topics of experimental designs,which include the constructions of space-filling designs with attractive stratifications,space-filling designs under maximin distance criteria,and screening designs in the case of sequential addition of experimental runs.Next we briefly introduce the motivations of this dissertation.With the rapid development of modern science and technology,some physical experiments are expensive and time consuming.Then an alternative is to make use of computer experiments rather than physical experiments.A significant feature is that computer experiments do not produce random errors,that is,the same inputs must produce the identical outputs.Therefore,the three fundamental principles of physical experiments,i.e.,replication,randomization,and blocking,are not necessary to consider in design and analysis for computer experiments.Since the research system of computer experiments is often extremely complicated,and indicates the existence of highly nonlinear relationship between the input variables and output responses.Then a desirable design is to scatter the design points over the experimental region as uniformly as possible,and is called a space-filling design.There exist some good criteria to evaluate the performance of space-filling designs,which include,e.g.stratifications,column-orthogonality,maximin distance and discrepancy criteria.First,some space-filling designs have the low-dimensional stratifications.Research in this area started from Latin hypercube designs(Mc Kay,Beckman and Conover,1979),continued in orthogonal array-based Latin hypercube designs(Owen,1992;Tang,1993),and became popular in strong orthogonal arrays(He and Tang,2013,2014;He,Cheng and Tang,2018)and mappable nearly orthogonal arrays(Mukerjee,Sun and Tang,2014).Besides,a desirable design should have columnorthogonality or near column-orthogonality,which guarantees the attractive uniformity in its two-dimensional projections(Bingham,Sitter and Tang,2009).For some recent developments of column-orthogonal strong orthogonal arrays,we refer to Liu and Liu(2015)and Zhou and Tang(2019).This dissertation will discuss the recent advances of such designs,and provide the constructions of column-orthogonal strong orthogonal arrays with high levels,column-orthogonal nearly strong orthogonal arrays,and other four new classes of mappable nearly orthogonal arrays.The space-filling designs with attractive stratifications have low-dimensional uniformity,and one of their disadvantages is that any such design usually does not satisfy full-dimensional space-filling property.Johnson,Moore and Ylvisaker(1990)introduced the maximin distance criterion to address this problem.This criterion seeks designs by maximizing the minimum distance between design points such that the design have an attractive full-dimensional space-filling property.Therefore,it is worth constructing maximin distance strong orthogonal arrays and maximin distance designs with the mappable nearly orthogonal array structure.Existing approaches to finding maximin distance designs often resort to algorithmic searches.When the numbers of runs and factors are large,the performance of such a search deteriorates.This shows that it is worth constructing maximin distance designs with attractive stratifications through some systematic approaches.Finally,we consider the construction of screening designs in the case of sequential addition of experimental runs,and its basic idea is the same as sequential designs in computer experiments,that is,given an initial design,we add another part of design points such that the whole design satisfies certain good properties.The commonly used screening designs are two-and three-level factorial designs,where three-level designs are beneficial to interaction detection.For three-level screening designs,one of the frequently used designs is the fractional factorial design,and the consequence is the aliasing of factorial effects.Then an important issue is how to add relatively small design points to the initial design such that the whole design can reduce the aliasing of factorial effects as much as possible.For two-level screening designs,it is worth considering the construction of factorial in the case of sequential addition of experimental runs.Specifically,the topic that deserves further study is how to construct compromise designs that achieve a bias-variance tradeoff.This dissertation is devoted to the above new issues.Now,let us introduce the main contents of each chapter briefly.Chapter 1 is the introduction,consisting of some backgrounds.Chapter 2 presents a construction method for column-orthogonal strong orthogonal arrays.This chapter proposes column-orthogonal strong orthogonal arrays of strength two star and three.Construction methods and characterizations of such designs are provided.The newly constructed designs,with the numbers of levels being increased,have their space-filling properties in one and two dimensions being strengthened.They can accommodate comparable or even larger numbers of factors than those in the existing literature,enjoy flexible run sizes,and possess the column orthogonality.The construction methods are convenient and flexible,and the resulting designs are good choices for computer experiments.Chapter 3 proposes two construction methods for column-orthogonal nearly strong orthogonal arrays.Strong orthogonal arrays enjoy more attractive space-filling properties than ordinary orthogonal arrays for computer experiments.This chapter proposes two methods for constructing column-orthogonal nearly strong orthogonal arrays via ordinary orthogonal arrays.The newly constructed designs enjoy column orthogonality,can accommodate twice or more number of factors than the existing strong orthogonal arrays,and most columns of these designs have the attractive two-dimensional space-filling property of strong orthogonal arrays.In addition,the proposed designs with four levels enjoy an attractive space-filling property under the maximin distance criterion.Chapter 4 provides some new methods to construct four new classes of mappable nearly orthogonal arrays.Motivated by the idea of mappable nearly orthogonal arrays,this chapter proposes some methods for constructing four new classes of spacefilling designs.Such designs enjoy attractive space-filling properties,column orthogonality between groups,and accommodate a large number of factors such that they are useful for computer experiments.The direct replacement method and the generalized doubling play key roles in these constructions,making construction methods simple and general.Chapter 5 provides a method for constructing maximin distance strong orthogonal arrays.The maximin distance designs is an attractive class of space-filling designs for computer experiments.Systematic construction of such designs is challenging.This chapter studies the space-filling properties of strong orthogonal arrays under the maximin distance criterion and proposes a method for constructing maximin distance(nearly)strong orthogonal arrays,where the level expansion rule based on orthogonal arrays is important in the construction.These designs enjoy better full-dimensional space-filling property than the existing strong orthogonal arrays.In addition,the nearly constructed strong orthogonal arrays can accommodate twice number of factors than the existing strong orthogonal arrays,and have higher distance efficiencies than the latter.Chapter 6 provides a method for constructing maximin distance designs.Recently,constructions of maximin distance designs often make use of highly specialized techniques.This chapter presents an easy-to-use method for constructing maximin distance designs.The proposed method is versatile as it is applicable for any distance measure.The method is based on the direct replacement method,that is,replacing the levels of a design by the rows of another design,and its basic idea is to construct large designs from small designs and the method is effective because the quality of large designs is guaranteed by that of small designs,as evaluated by the maximin distance criterion.Chapter 7 provides semifoldover designs for three-level orthogonal arrays with quantitative factors.With the use of linear-quadratic system,this chapter considers semifoldover designs for three-level orthogonal arrays with quantitative factors.We examine when the linear effects can be de-aliased from their aliased two-factor interactions for regular and nonregular designs,and obtain some good properties via semifolding over on partial factors or all factors.Theoretical results and some examples are provided to illustrate the usefulness of the proposed designs.Chapter 8 provides a systematic construction of compromise designs under baseline parameterization.This chapter focuses on the estimation of main effects for two-level factorial designs under the baseline parameterization.Compromise designs are recently introduced as an attractive class of designs that achieves the trade-off between the bias and the variance for estimating main effects.This class of designs is mostly based on algorithmic search,while its systematic construction is rarely studied.This chapter provides a systematic construction method for a new class of compromise designs.The closed-form expressions of the newly constructed designs are established in terms of a bias and efficiency criteria.Some theoretical results and empirical comparisons with the existing designs are provided to show the usefulness of this class of designs.Chapter 9 concludes this dissertation with some discussions.