The Study Of Uniform Property Of Several Classes Of Complex Designs

Science and technology have come to be an important motivation to develop a country constantly,especially,high technology is embraced exclusively over all countries.Computer experiment has been an efficient tool to simulate complex systems,and plays a significant role in the research and development of high-tech products,such as chemical industry,manufacturing industry and pharmaceutical industry,see Snatner,et al.(2003)and Fang,et al.(2006).In a complex system,it is always difficult to obtain an analytical formula between inputs and output,because the relationship between them may be linear or nonlinear,and is even unknown.However,we can choose an approximate analytical expression,also called as metamodel,to connect the output with inputs.With the help of field knowledge and some data sets through a computer simulations under some selected experiment points,we can make deep insight of the relation between inputs and output for the considered system.When designing a computer experiment,one needs to take care that the ex-periment points should provide the information about different areas of the given experimental region.In other words,experiment points should be scattered evenly over the experimental region,which is called as space-filling property.There are three popular ways to evaluate how to scatter evenly these experimental points in literature:one way considers the uniformity of one-dimension or low-dimension of designs,such as Latin hypercube design and its modification,see Mckay,et al.(1979)and Tang(1992);another way is measured by separating distance and filling distance,which results in maximin distance design and minimax distance designs,see Johnson,et al.(1990);the third way depicts the uniformity of designs by the deviation between the empirical distribution of experimental points contained in a design and the uniform distribution over the experimental region,see Fang,et al.(2006).In order to study complex systems,sequential computer experiment has recently validated to be a useful strategy.Ranjan,et al.(2008)and Xiong,et al.(2013)model the sequential computer experiment by modifying the convenient Gaussian process model.Xu,et al.(2015)suggested sequentially refined Latin hypercube de-sign for designing sequential computer experiment.However,different sequentially refined Latin hypercube designs may be possess different uniformity,and sometimes their uniformity may be bad.Therefore,it is worthy explored how to make sense of the space-filling property for sequential computer experiment.The uniformity measured by discrepancy has not only been successfully applied into experiment field,but also has been useful for quasi-monte carlo methods.More importantly,the measure based on discrepancy can be used to select good Latin hypercube de-signs and traditional factorial designs,see Fang and Mukerjee(2000),Fang,et al.(2001)and Fang,et al.(2002).Recently,Butler and Ramos(2007)and Gupta,et al.(2010)considered how to add optimally some new runs into known orthogonal arrays and supersaturated designs based on different criteria,respectively.If we adopt uniformity criterion based on discrepancy to describe the evenness of sequen-tial experiment,then these three kinds of designs can be integrated into a unify framework,which will bring a scientific guidance to the practioners for selecting the good designs.A complex system always contains a lot of input variables,also called as fac-tors,which may affect the output of the system with varying extents.Therefore,experimenters need to identify the factors which have significant effect such that the experimenters will consider the projective uniformity of designs.Fang and Qin(2005)and Chatterjee,et al.(2012)introduced the uniformity pattern to study the projective uniformity of designs,but they only focus on the two-level designs or de-signs with qualitative factors.In the application of computer experiment,sometimes experimenters want to study the system more precisely,then the high-level quanti-tative designs are required.Besides,the inputs of system may include qualitative factors and quantitative factors simultaneously,which motivates a lot of researches about modeling and designing of such experiments,see Qian,et al.(2008)and Deng,et al.(2015).It is an important issue to consider the space-filling property of this kind of designs,and which is also helpful for practioners to select good designs.It has been of interest to construct designs with good uniformity for theoreti-cal researchers and praetioners.There are some methods for constructing uniform designs in literature,such as the method based on special combinatorial structures,the method based on heuristic algorithm for searching and permuting the levels of "good" designs.Generally,constructing uniform designs with low-level is easier than that with high-level.The level combination replacement method establishes a bridge between designs with low-level and high-level,and has been successfully applied into constructing high-level orthogonal arrays based on low-level orthogonal arrays,see Hedayat,et al.(1999).Fang and Mukerjee(2000)pointed out that there is a close connection between uniformity and orthogonality.Therefore,it is rational to expect that level combination replacement may construct high-level designs with good uniformity.This dissertation contains the following nine chapters.Chapter One is intended to be an introduction.It gives a comprehensive review and analysis for background and motivation of this dissertation,proposed some problems which will be explored in this dissertation,and describes the structure and innovations of this dissertation.Some necessary materials are also shown,which help go through and understand the work of this dissertation.Chapter Two studies the issue of uniformity of mixed two and three-level extended designs.Following the characteristic of follow-up experiments,a new class of designs are introduced,called as extended designs,which are obtained by adding some new experimental points into a selected designs with good property.This chapter will study the uniformity of mixed two and three-level extended designs in terms of wrap-around L2 and Lee discrepancies.In view of uniformity to mea-sure the goodness of extended designs,the following two cases can be integrated into a whole uniformity framework:one is the goodness of designs obtained by adding some new runs into two-level orthogonal arrays studied by Butler and Ramos(2007),and another is how to evaluate the goodness of adding some new runs into a given supersaturated design reported by Gupta,et al.(2010).This chapter also estab-lishes a connection between the discrepancy of extended designs with its initial and additional designs,which brings insight into constructing uniform extended designs for application.With the help of the vector of frequency and coincidence number,lower bounds of wrap-around L2 and Lee discrepancies of extended designs are also provided,respectively.Chapter Three considers the issue of uniformity of extended designs with high levels.Extended designs with high-level give a broader alternatives to computer experiments.Under the generalized discrete discrepancy,this chapter studies the uniformity of general extended designs,and establishes a discrepancy connection among the extended design,its initial and additional designs.According to the vector of Hamming distance,a lower bound of discrepancy of general extended designs is also given,which provides a useful benchmark to select uniform extended designs for practical application.Chapter Four studies the issue of projective uniformity of two-level extended designs.When the experiment contains a lot of factors,it is often happen that not all of factors bring a significant influence into the output of the system.Therefore,one should identify the factors which possess significant influ-ence.It needs to consider the projective uniformity of extended designs.Uniformity pattern is an important quantity to evaluate the projective uniformity.Under the centered L2 discrepancy,this chapter studies the uniformity pattern of extended designs,and establishes a connection of uniformity pattern among the extended de-sign,its initial and additional designs.And a lower bound of uniformity pattern of extended designs is also obtained,which provides a benchmark to compare the projective uniformity of different extended designs.Chapter Five works out the projective uniformity of general high-level designs with quantitative factors.The continuous variables with high-level are often encountered in computer experiment,and it is always occurred that each variable has different influence.In order to consider the cost,the factors with significant effect will be took more values than that with less influence.Moreover,the experimenter should also identify the factors which have significant effect.Under wrap-around L2 discrepancy,this chapter studies the uniformity pattern of designs with quantitative factors,and provides a connection between these two ways to define uniformity pattern.An analytical formula of uniformity pattern is established,which is convenient to select minimum projective uniform design based on uniformity pattern for applications.a lower bound of uniformity pattern under wrap-around L2 discrepancy is also provided.Chapter Six considers the issue of constructing minimum projective uniform designs with quantitative factors.Cheng and Wu(2001)pointed out that level permutation can change the geometrical structure of designs and their statistical property.Following this result,this chapter studies the uniformity pattern of quantitative factor designs in terms of level permutation.Under wrap-around L2 discrepancy,an analytical linear expression between the average uniformity pattern and generalized wordlength pattern,and between the average uniformity pattern and orthogonality,is respectively obtained,which reveals that permutating the level of traditional minimum generalized aberration designs or orthogonal arrays can result in designs with good projection uniformity.This chapter also establishes a lower bound of average uniformity pattern,which provides a good stopping criterion to search designs with good projection uniformity by heuristic algorithm.Chapter Seven studies the issue of uniformity designs with qualitative and quantitative factors simultaneously.When computer experiment contains qualitative and quantitative factors simultaneously,how to design such an exper-iment becomes to be meaningful.Although Qian(2012)and Deng,et al.(2015)proposed sliced Latin hypercube designs and its modification to conduct this kind of experiments,but these designs may have bad uniformity.In order to evaluate the uniformity of this kind of designs,the chapter introduces a new discrepancy,called as marginal coupled discrepancy(MCD for short),which not only considers the different nature of qualitative and quantitative factors,but also makes a balance between the local uniformity and global uniformity.A sound statistical justification of MCD is also given,which is a connection between average MCD with the con-sider of level permutation and generalized wordlength pattern.And a lower bound of MCD is also given.Chapter Eight studies the uniformity of four-level designs constructed from two-level designs.Hedayat,et al.(1999)had successfully adopted level combination replacement to construct high-level orthogonal arrays from low-level orthogonal arrays.Further,Phoa and Xu(2009)applied level combination replace-ment to obtain the two-level minimum generalized aberration designs.There is a close connection among the uniformity,aberration and orthogonality.For example,Fang,et al.(2000)and Fang and Mukerjee(2000)pointed out that orthogonal ar-rays and minimum aberration design tend to possess good uniformity.Therefore,it is ration to expect that the four-level designs constructed by level combination replacement from two-level designs will perform well under uniformity criterion.Un-der mixture discrepancy,this chapter establishes an analytical relationship between the constructed four-level designs and its two-level cornerstone designs,which sup-ports the above expectation theoretically.A connection between the aberration of two-level cornerstone designs and the uniformity of four-level designs constructed is given.Applying the doubling technique,the limitation of level combination re-placement is overcame,which broadens the range of valid cornerstone designs.The performance of level combination replacement with the help of doubling technique is deeply studied.This chapter obtains an analytical expression of mixture dis-crepancy between the double design and its initial design,and also establishes an analytical relation between the mixture discrepancy of double design and aberration of its initial design.A lower bound of alternative mixture discrepancy measure for four-level designs constructed is also provided.Chapter Nine gives a brief conclusion and further studies of this dissertation.