Theory And Application Of Augmented Space-filling Designs

An experimental design can explore the relationship between the factors and response by rationally arranging the level combination of factors and the number of experimental points.When the model between the response and factors is unknown,a space-filling design is suitable.As a process for exploring unknown problems,we may choose a design with small number of runs in the initial stage.After analyzing the data of the initial design,we may find that the existing data is not enough to achieve the intended purpose and need a follow-up design.A design of adding experimental points sequentially is called the sequential experimental design.Most of the existing literatures on sequential experimental designs only add experimental points to the initial design without changing the number of levels of factors.However,in many practical problems,the levels of some factors need to be extended in follow-up stage to meet the requirement of the entire experimental analysis.We call such a design as the levelaugmented design.At present,there are few studies on level-augmented designs,and the research in this thesis plays an important role in solving practical problems.According to the levels of factors,low-level level-augmented designs are suitable for physical experiments,and high-level sequential uniform designs are suitable for computer experiments.The contents of this thesis are as follows:1.Level-augmented uniform designsWe firstly give the definition of level-augmented designs,which can be divided into range-extended level-augmented designs（RELADs）and range-fixed level-augmented designs（RFLADs）,according to whether the experimental domain is extended or not.We use the wrap-around L₂-discrepancy（WD）to measure the uniformity of those levelaugmented designs.The expressions of the squared WD-value for those level-augmented designs are derived.The lower bounds of WD for range-extended and range-fixed level-augmented designs are also obtained under some special parameters,respectively.Moreover,we present a method to construct one kind of level-augmented designs where the parameters satisfy some conditions,and prove that the constructed levelaugmented designs are the level-augmented uniform designs when the parameters meet the certain conditions.Finally,an example is given to illustrate the effectiveness of level-augmented uniform designs.2.Maximin L1-distance range-fixed level-augmented designsSince mixed two-and three-level range-fixed level-augmented designs and mixed two-and four-level range-fixed level-augmented designs are widely used in practice,we consider these two special kinds of RFLADs.We use the maximin L1-distance criterion to measure the space-filling property of these two types of RFLADs,and give the corresponding tight upper bounds of L1-distance.We give some construction methods for the two kinds of RFLADs,and prove that when the parameters meet the certain conditions,the constructed designs are the maximin L1-distance RFLADs.Moreover,we also discuss the column-orthogonality of mixed two-and four-level RFLADs.3.High-level sequential uniform designsMost of level-augmented designs are added points sequentially to low-level initial designs,the additional number of points may be the minimal value that makes the resulting level-augmented designs to be U-type,and the number of level augmentation of factors is not large.Therefore,those level-augmented designs are suitable for physical experiments.However,in many practical applications,the number of levels of factors is large,and we need to add multiple stages of experimental points sequentially to a high-level initial design.Therefore,we further study high-level sequential uniform designs.We use a good lattice point set as the initial design,and add experimental points by multiple stages based on the good lattice point set,and ensure that the sequential design points obtained on each stage are evenly distributed in the experimental area.According to whether the experimental domain is fixed or not,we give the construction methods of adding points sequentially on the fixed experimental area and on the contracted experimental area,respectively.We also give a generation algorithm for sequential uniform designs,which can be used to solve optimization problems.We illustrate the effectiveness of the algorithm by using some examples.The various types of level-augmented designs and high-level sequential uniform designs studied in this thesis belong to the aspect of augmented space-filling designs.They are useful for solving practical problems.