Design And Analysis For Sequential Experiments And Order-of-Addition Experiments

As a tool for studying all kinds of complex processes and systems,designs of experiments are widely applied in scientific research,industry,agriculture,medicine and other fields.Designs of the experiment can be divided into single designs and sequential designs.A single design is to complete a fixed number of experiments at one time,while a sequential design sequentially adds design points until a given experimental goal is reached.As a cost-effective method,sequential designs can avoid waste caused by blindly increasing the number of experiments,and can not fail to come up with a conclusion due to too few experiments.With the rapid development of computer technology,the existing sequential designs are insufficient to meet the requirements of multiple complicated systems.Therefore,experimenters urgently need to investigate the theories and construction methods of a broad range of sequential designs.This dissertation mainly focuses on the designs and analyses of the following three types of complex sequential experiments,i.e.,robust composite designs,sequential Latin hypercube designs in a variety of changing spaces,and the homogeneity of variance test and robust parameter analysis of order-of-addition experiments.For these three types of sequential designs and their analyses,some have partial research results,and some have not yet.Therefore,this dissertation conducts an in-depth study on the design and analysis of these three types of complex sequential experiments.Below we briefly describe the research background and current research results of these three types of complex sequential designs and the related analyses.As a special type of sequential designs,composite designs are widely applied in the response surface methodology.The response surface methodology proposed by Box and Wilson(1951)is widely used in input-output systems that explore unknown relationships.Based on the second-order model,commonly used composite designs include central composite design(CCD,Box and Wilson,1951),small composite design(Draper and Lin,1990),subset design(Gilmour,2006),augmented pairs design(Morris,2000),orthogonal-array composite design(OACD,Xu,Jaynes and Ding,2014)and definitive screening composite design(DSCD,Zhou and Xu,2017).Compared with other composite designs,OACD has higher estimation efficiency and the ability for multiple analysis for cross validation.However,the number of runs is often high,especially for high-dimensional cases.In addition,when the second-order model has misspecification,the existing composite designs are not robust.Therefore,it is necessary to find a type of robust composite design with flexible numbers of runs and factors.As an economic and efficient method,sequential designs are widely used in computer experiments.Latin hypercube design(LHD,Mc Kay,Beckman and Conover,1979)is one of the commonly used designs for computer experiments.For an LHD,each level of each variable appears only once.For computer experiments with different accuracies,Qian and Wu(2008)proposed a class of nested designs in the invariant space.Subsequently,Qian,Tang and Wu(2009),Qian,Ai and Wu(2009)and Sun,Liu and Qian(2014)used algebraic methods to construct some special types of nested space-filling designs.Rennen et al.(2010)and Chen and Xiong(2017)used algorithms to construct space-filling nested Latin hypercube designs(NLHDs).These methods are either only feasible for certain numbers of runs or factors,or the computational complexity is too high to be affordable.In addition,the existing NLHDs are all constructed in invariant spaces.For NLHDs in space contractions,these designs are no longer applicable.Therefore,how to construct flexible,simple,and space-filling NLHDs in a wide variety of experimental spaces is worth studying.Sequential designs can also be used to complete the testing of heteroscedasticity and robust parameter analysis.In the past few decades,the order-of-addition(Oof A)experiment has received widespread attention.The response of this experiment is often related to the order in which the different components are added to a system.However,the existing research results only consider the Oof A experiments of the same variance.For Oof A experiments with heteroscedasticity,there is still no research result.The following is the structure of this dissertation.Chapter 1 introduces the research background and current research results of sequential experiments.In addition,this chapter also provides some necessary preliminary knowledge.Chapter 2 proposes a new type of composite designs and provides its construction method.This composite design is called orthogonal uniform composite design(OUCD),which combines an orthogonal array and a uniform design.This type of composite designs is more robust than other types of composite designs under certain conditions.Moreover,some construction methods for OUCDs under the maximin distance criterion are provided and their properties are also studied.Furthermore,such designs not only maintain the advantages of OACDs such as high estimation efficiencies and the ability for multiple analysis for cross validation but also have more flexible run sizes than CCDs and OACDs.Chapter 3 proposes a new type of sequential LHDs(SLHDs)via good lattice point(GLP)sets and provides its construction method.This SLHD is called the sequential GLP(SGLP)sets,with the flexibility in the run size and the number of factors.Moreover,this chapter provides fast and efficient approaches for identifying spacefilling SGLP sets under a given criterion.Furthermore,combined with SGLP sets and the linear level permutation technique,a class of asymptotically optimal maximin distance SLHDs via GLP sets for two space varieties is obtained where the L1-distance of each layer is optimal or asymptotically optimal.Numerical results demonstrate that SGLP sets have better space-filling properties than the existing SLHDs in the invariant space.Besides,the space-filling SGLP set has less computational complexity and more adaptability.Chapter 4 investigates the homogeneity of variance test and robust parameter analysis of the order-of-addition(Oof A)experiment.It tests the homogeneity of replicate Oof A experiments first.If the variance is heteroscedastic,experimenters conduct sequential designs of Oof A for dual response models and their robust parameter analyses.Based on the pair-wise-order(PWO)model,the obtained orders from the proposed methodologies not only achieve the goal of the response mean,but also minimize the standard deviation within the framework of Oof A experiments.Simulation studies are used to illustrate these methodologies.It is shown that the proposed methods perform well for replicated Oof A experiments.Chapter 5 proposes another general approach for solving the dual response problem for Oof A experiments involving environmental(noise)variables and heteroscedasticity.Moreover,it provides a robust and simulation-based optimization method accounting for collected(historical)data on the uncontrollable environmental variables and obtains the robust orders.Furthermore,some efficient approaches are presented to solve the corresponding discrete optimization.Theoretical supports are given under specific setups.Some numerical simulations illustrate the effectiveness of the proposed approach for dual response optimization of Oof A experiments.Chapter 6 concludes this dissertation.