Construction Of Complex Computer Experimental Designs And Nonregular Designs

Experiments are ubiquitous in the fields of industry,agriculture,engineering,and science.In general,experiments can be divided into two classes: physical experiments and computer experiments.In physical experiments,scientists conduct laboratory experiments or make field observations.As there always exist random errors in physical experiments,the experimenter might obtain different output responses with the identical input variables.Existence of random errors creates complexity in data analysis and modeling.To solve this problem,physical experiments always follow the following three fundamental design principles,i.e.randomization,replication,and blocking.When the physical experiments are costly in terms of budget or time,or it is dangerous to conduct certain physical experiments in the actual environment,we can explore the relationship between input variables and output responses with the help of a computer experiment.Examples include the assessment of the long-term effects of climate change and the car crash tests.An outstanding feature of computer experiments is that the same input variables always produce the identical output responses.Correspondingly,the design and analysis of computer experiments differ a lot compared to those of physical experiments.First,the three fundamental design principles of physical experiments,i.e.randomization,replication,and blocking,need not be considered in computer experiments.Second,unlike physical experiments,it is easy to change the levels of factors in computer experiments and,thus,high-level factors are entirely possible for computer experiments.The true relationship between input variables and output responses is always complex.An important goal of computer experiments is to obtain an approximate model that is relatively simpler than the real model and can be used as a good surrogate for the real model.Here the approximate model is also called a metamodel.Suppose the metamodel is a polynomial model or a Gaussian process model,then the designs with column-orthogonality are preferred since such a design allows uncorrelated estimations of the main effects in the polynomial model and effective factor screening in the Gaussian process model.When the metamodel is unknown,space-filling designs are the best choice for computer experiments and they are robust to model misspecification.In summary,column-orthogonality and space-filling property are two desirable criteria for finding good designs in computer experiments.Numerous approaches have been proposed in literature for constructing space-filling designs and column-orthogonal designs separately,yet few works have examined on designs with both properties.Recently,He and Tang(2013)introduced strong orthogonal arrays(SOAs)and He,Cheng and Tang(2018)presented SOAs of strength two plus for economic reason.Liu and Liu(2015)and Zhou and Tang(2019)studied the column-orthogonal SOAs.Besides,Mukerjee,Sun and Tang(2014)proposed mappable nearly orthogonal arrays.Note that both column-orthogonal SOAs of strength two plus and mappable nearly orthogonal arrays perform well in both two-dimensional space-filling properties and column-orthogonality.These arrays,however,can accommodate too few columns and the run sizes of mappable nearly orthogonal arrays are not flexible,which limit their use in practical applications.A research direction worth pursuing is to construct designs that can accommodate more columns and still enjoy the comparable desirable space-filling properties and column-orthogonality.To achieve the same space-filling properties in three dimensions as ordinary orthogonal arrays,the strength of SOAs should be three or higher.Shi and Tang(2020)introduced a new class of SOAs of strength three which enjoy almost all of the spacefilling properties of SOAs of strength four,and have much more columns than the latter ones.However they did not take the column-orthogonality into consideration.The construction of these kind of SOAs with column-orthogonality is worth considering.Based on the aforementioned arrays,we can construct designs with desirable spacefilling properties in low dimensions for computer experiments.In addition,orthogonal arrays can also be used to construct space-filing designs.Tang(1993)constructed Latin hypercube designs from orthogonal arrays by level expansion.Among the orthogonal arrays,two-level regular fractional factorial designs are commonly used for screening experiments,since they are easily interpreted.For these regular designs,factorial effects are either orthogonal or fully aliased.In contrast,a nonregular fractional factorial design is one for which some factorial effects are partially aliased.Compared with regular fractions,nonregular designs have more complicated aliasing structure,but they are more flexible in run sizes and allow estimation of more effects.Parallel flats designs proposed by Connor and Young(1961)are popular nonregular designs because they retain some of the simplicity of regular fractional factorial designs and have flexible operating dimensions.Most of the work on parallel flats designs focuses on the parallel flats designs with three flats.Generalizing to parallel flats designs with f flats(f is larger than 3)is more challenging.This dissertation will focus on the above new issues.In the following,let us introduce the contents of each chapter briefly.Chapter 1 is the introduction,consisting of some backgrounds,concepts and notations that will be used in the following chapters.Chapter 2 presents several new approaches to construct space-filling orthogonal designs.This chapter explores the construction of a new type of design,which includes the orthogonal Latin hypercube design as a special case.These designs are not only column-orthogonal but also have good space-filling properties in low dimensions,and can accommodate large number of factors.All these appealing properties make them good choices for designing computer experiments.Based on orthogonal arrays,the proposed methods are easy to operate and flexible in run sizes.Many new orthogonal designs with desirable space-filling properties are constructed and tabulated.Rotation matrices play a key role in the construction.Chapter 3 proposes the strong-group orthogonal arrays and provides the construction methods.This chapter proposes a new class of designs called strong grouporthogonal arrays whose columns can be partitioned into groups with the columns from different groups being column-orthogonal and enjoying attractive space-filling properties in low dimensions.Meanwhile,the whole arrays can be collapsed to fully orthogonal arrays which accommodate large numbers of factors.Methods for constructing this class of arrays based on both regular and nonregular designs are proposed.Difference schemes play a key role in the construction.Chapter 4 presents the method for constructing column-orthogonal strong orthogonal arrays of strength plus.Shi and Tang(2020)introduced a new class of SOAs of strength three which enjoy almost all of the space-filling properties of SOAs of strength four,and have much more columns than the latter ones.In this chapter,we call this class of arrays as SOAs of strength three plus.This chapter proposes a method for constructing column-orthogonal SOAs of strength three plus.The resulting designs are more advantageous than the designs in the Shi and Tang(2020)and can accommodate the same number of columns.In addition,by juxtaposing several columnorthogonal SOAs of strength three plus,we get a series of grouped designs,where each group has good column-orthogonality and space filling properties in both two and three dimensions,and the whole arrays can accommodate a large number of columns.This type of grouped design is very useful for the screening experiments in the initial stages of experiments.Chapter 5 explores the general theory of two-level parallel plane designs.This chapter aims to study the general theory for the parallel flats designs with f flats for any f > 3.We propose a method for obtaining the confounding frequency vectors for all nonequivalent parallel flats designs,and to find the least G-aberration(or highest Defficiency)parallel flats design constructed from any single flat.We also characterize the quaternary code design series as parallel flats designs.Parallel flats designs are particularly useful for constructing nonregular fraction,split-plot or randomized block designs.Finally,we show how designs constructed by concatenating regular fractions from different families may also have a parallel flats structure.Chapter 6 concludes the work of this dissertation and provide some discussions.