Construction Of Optimal Supersaturated Designs And Orthogonal Designs

Design of Experiment,a branch of statistics, has enjoyed a long history of the-oretic development as well as application.It is widely used in experimental investi-gations,and statistically designed experiments have been a key tool for improving quality in industry and manufacturing.As science and technology have advanced to a higher level,investigators are becoming more interested in and capable of studying large-scale systems for this area. To address the challenges posed by this techno-logical trend,research in experimental design has lately focused on the class of supersaturated design(SSD) and non-regular orthogonal design.SSDs have become increasingly popular in recent years because of their potential for saving run sizes and their technical novelty. SSD is a form of factorial design in which the degrees of freedom for all its main effects exceeds the total number of distinct factorial level-combinations of the design. The main reasons for considering SSDs is run size economy. Based on the effect sparsity principle,we can utilize SSDs for screening significant factors.Meanwhile,orthogonal design plays an important role in the design of experi-ment.Orthogonality here implies that for every two columns all the possible level-combinations occur with the same frequency, and thus an orthogonal design is in fact an orthogonal array with strength at least two.Orthogonal factorial designs can be broadly classified into two categories:regular orthogonal designs and non-regular orthogonal designs.A regular orthogonal design is determined by its defining rela-tions and has a simple aliasing structure in that any two effects are either orthogonal or fully aliased. In contrast,a non-regular orthogonal design exhibit some complex aliasing structure, meaning that there exist effects that are neither orthogonal nor fully aliased.Extensive researches have been done on SSDs and non-regular orthogonal designs in recent decades,with main focuses on the optimality theory and design construc-tion. Several new kinds of criteria have been proposed for assessing them.Among these criteria, Fang,Lin and Liu(2003)proposed the E(fNOD)criterion which mea-sures the two-factor non-orthogonality of a design averagely and Xu(2003)intro-duced the minimum moment aberration (MMA)which investigates the relationship between runs,instead of studying the relationship between factors.The MMA is conceptually cheap and has tremendous computational advantages.So we use MMA to evaluate the orthogonal designs in this dissertation.Actually, the E(fNOD) criterion and MMA are shown to be equivalent for symmetrical designs in many references.While, neither of them is enough to prevent the existence of fully aliased columns in designs.So we also use the maximum fNODij criterion to further discrim-inate the SSDs.From their definitions and concepts,it is more convenient to use the maximum fNODij and the E(fNOD) criteria together than the maximum fNODij and MMA together for assessing SSDs.So we mainly adopt the E(fNOD) and maximum fNODij criteria to assess SSDs in this dissertation.For two-level SSDs,the construction and analysis have been studied by many researchers,but there are only a few studies on the construction of optimal multi-level and mixed-level SSDs.A few multi-level and mixed-level optimal SSDs with certain parameters(run-numbers,column-numbers,level-numbers)have been con-structed in these studies,there are still a large number of multi-level and mixed-level SSDs need to be constructed for practical use.One of the main purposes of this dissertation is to investigate the construction of E(fNOD)-optimal multi-level and mixed-level SSDs.Cyclic generator is widely used to construct different kinds of designs in the literature.Plackett and Burman(1946)firstly introduced the cyclic designs to construct orthogonal saturated designs which include the two-level designs with run- numbers being not prime powers of two and the symmetrical multi-level designs with prime power run-numbers and prime level-numbers.We propose a new method to construct multi-level cyclic SSDs which can be regarded as a generalization of their method. Based on the knowledge of Galois field,we obtain the generator vectors for some cyclic (fNOD)-optimal designs with prime power run-numbers and prime power level-numbers.Some of these E(fNOD)-optimal designs are verified to be also optimal under the maximum fNODij criterion. The method we proposed is easy to manipulate and much more efficient than the searching method which are used to get the generator vector for some cyclic E(fNOD)-optimal multi-level SSDs at present. Some of the generator vectors that can produce cyclic E(fNOD)-optimal multi-level SSDs with run-numbers less than 100 are tabulated.Fang,Lin and Liu(2003)showed that if the coincidence number between any two distinct rows of a design is a constant,then it is E(fNOD)-optimal.From this,we obtain that if the coincidence number between any two distinct rows of a saturated balanced design is a constant,then it is E(fNOD)-optimal and E(fNOD)=0,i.e.it is an orthogonal design.This result immediately leads to the orthogonality of the Plackett and Burman designs with prime power run-numbers whose previous proof is much more difficult and complex.Furthermore,we explore how to construct E(fNOD)-optimal mixed-level SSDs usingκ-cyclic generators.The necessary and sufficient condition for the existence of mixed-levelκ-circulant SSDs with equal occurrence property are provided.These conditions help us to select thoseκ-cyclic generator vectors which can produce bal-anced SSDs from a huge number of candidate vectors.We also present some par-ticular properties of the mixed-levelκ-circulant SSDs constructed by this method. One of them is the sufficient condition under which the generator vector produces an E(fNOD)-optimal SSD.From it,we can easily judge whether a mixed-levelκ-circulant SSD is E(fNOD)-optimal,instead of checking all the coincidence numbers between distinct rows we only need to check the coincidence numbers between those rows which have different run number differences (if the sum of the absolute of two differences equals run number minus 1,they are regarded to be the same).The other property is that some of the fNODij's of aκ-circulant design are equal.These properties simplify the investigation of these designs a lot.The algorithm for finding generators of E(fNOD)-optimal designs is provided in detail. Many new E(fNOD)-optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean(2004)for constructing two-level SSDs and the one due to Georgiou and Koukouvinos(2006)for the multi-level case.The approach for constructing orthogonal designs from the Kronecker sums of difference matrices and orthogonal designs is widely known.Inspired by it,we have discussed the relationship between the coincidence numbers of the design being the Kronecker sum of a, balanced design and the transpose of a difference matrix and that of the balanced design.Based on this result,we propose a method to construct an E(fNOD)-optimal SSD by juxtaposing designs,one of which is the Kronecker sum of an E(fNOD)-optimal design and a difference matrix defined on an abelian group, and the others are the general Kronecker sums of the columns of an E(fNOD)-optimal design and some given columns,respectively. As we know,general Hadamard matrices are those special difference matrices whose transposes are also difference matrices.We present that if the difference matrix in our construction is a general Hadamard matrix, then the E(fNOD)-optimal design we obtained has no fully aliased columns.Many newly constructed SSDs are tabulated for practical use.As for the orthogonal arrays,the construction is well developed.And the criteria to evaluate orthogonal designs are proposed by many researchers.But there are only a few results on the construction of optimal orthogonal design.In addition,most of them focused on the construction of optimal regular designs.While,there has been increasing interest in the study of non-regular orthogonal designs because they enjoy some good projection properties and have run size flexibility. The construc-tion of optimal orthogonal designs among all the regular and non-regular designs remains challenging especially when the run size is large.So another purpose of this dissertation is to investigate the construction of orthogonal designs with MMA. We show that if the coincidence numbers between distinct rows of an orthogonal design take at most three values,one of which is the minimum possible coinci-dence number among the class of orthogonal designs with the same sizes and the other two are consecutive integers, then the orthogonal design has MMA.Based on this,we present several results which show that the Kronecker sum of some general Hadamard matrices and orthogonal designs are the orthogonal designs with MMA. Butler(2003b,2005)obtained some orthogonal designs with generalized minimum aberration by projecting specific saturated orthogonal arrays.Fang,Zhang and Li (2007)and Sun,Liu and Hao (2009) proposed some algorithms to construct designs with generalized minimum aberration.Some of the designs they obtained are the same as ours.However,for large run-numbers,it is not so easy to find the specific saturated orthogonal arrays and the searching methods are apparently not efficient. Comparing with these methods,some of our construction methods work well even when the run number is very large.