四水平计算机试验设计的构造

编码理论在最优设计的构造中已被广泛应用.本文通过修正Gray映射编码变换来构造用于计算机试验的四水平均匀设计,其效率通过离散偏差进行刻画;同时也给出了离散偏差的下界,这个下界可作为获得最优设计的一个基准.     

可分解不完全区组设计的若干结果

考虑某类可分解不完全区组设计——PRIB设计的均匀性. 在一离散偏差度量意义下, 得到了PRIB设计是最均匀设计的一个充要条件；证明了均匀PRIB设计是关联的；提出了用某类U-型设计构造这类区组设计的方法, 并给出了该类设计的一个存在性结果, 这一方法建立起了PRIB设计和U-型设计之间的一重要桥梁.     

混合偏差下多水平的均匀列扩展设计

列扩展设计是添加试验因素,并安排跟随试验的一个重要方法.本文分别构造了四水平和五水平的列扩展设计,基于平均混合偏差准则研究了该类设计的均匀性,并得到了四水平和五水平列扩展设计的平均混合偏差的一个新的下界.数值例子表明本文中构造的列扩展设计具有非常高的效率.     

U-型设计的对称化L2-偏差的下界

均匀设计是部分因子设计的主要方法之一,已被广泛地应用于工业生产、系统工程、制药及其他自然科学中.各种偏差被用来度量部分因子设计的均匀性,其关键的问题是寻找一个精确的偏差下界,因为它可以作为衡量设计均匀性的标准.该文给出了4水平对称U-型设计的对称化L2-偏差的下界,以及2、3混水平和2、4混水平非对称U-型设计的对称化L2-偏差的下界.     

三水平U-型设计在对称化L_(2-)偏差下的下界

均匀试验设计是部分因子设计的主要方法之,已被广泛地应用于工业生产、系统工程、制药及其他自然科学中.各种偏差被用来度量部分因子设计的均匀性.不管使用哪种偏差,关键的问题是寻找一个精确的偏差下界,因为它可以作为衡量设计均匀性的标准.本文应用条件极值的方法得到了三水平U-设计在对称化L_(2-)偏差下的下界,该下界可作为寻找均匀设计的一个基准.     

平均混合偏差下混水平设计的均匀性

基于因子的水平置换方法讨论了混水平设计在平均混合偏差准则下的均匀性,得到了平均混合偏差的一个新下界,并研究了其所有水平置换设计的平均混合偏差与广义字长型之间的解析联系.从数值结果上表明该下界是可达的,且利用投影的方式可获得平均混合偏差下高效率的混水平设计.     

低偏差OALHD的构造

本文给出了利用均匀设计和正交表构造低偏差OALH设计的方法，该方法构造的设计既有优良的均匀性具有正交设计的均衡性，一个更重要的优点是可以构造较大样本容量的设计点集，本文同时给出了某些参数的均匀设计表，这些设计优于现有的均匀设计，具有实用价值。     

两水平U-型设计的扩大设计在投影加权对称偏差下的均匀性

均匀设计以其稳健和使用方便、灵活的特性而广受欢迎.为获得实验目标区域内散布均匀的设计点集,不同的均匀度量标准相继被提出.目前被广泛应用的有中心化L₂-偏差、可卷型L₂-偏差、混合偏差等.对称化L₂-偏差具有更好的几何性质,但受限于投影均匀性差的缺陷,使用范围十分有限.为了改进对称化L₂-偏差的低维投影均匀性,基于指数加权方式的投影加权对称化L₂-偏差的概念被提出,加权后的对称化L₂-偏差既能保留原偏差的各种优良性质,同时有效克服原来的缺陷并有更优异的表现.折叠翻转是构造因子设计时非常有用的技巧.本文利用投影加权对称偏差来作为评价折叠翻转方案的最优性准则,得到了两水平U-型设计在一般折叠翻转方案下扩大设计的投影加权对称偏差的下界,该下界可以作为寻找最优折叠翻转方案的基准.     

Lee偏差下二三混水平因子设计的最优添加

根据序贯试验通过一个阶段试验接着另一个阶段试验不断扩充的特征,新的试验点将会被添加到已经选好的设计中,因此,如何设计一个好的序贯试验是一个非常有意义的问题.在Lee偏差度量下,本文研究均匀的非对称拓展设计,并用它来做序贯试验.本文建立二三混水平拓展设计的Lee偏差与其初始设计和附加设计的Lee偏差之间的解析关系,为构造(近似)均匀的拓展设计提供了理论支撑.为了给出筛选、评价拓展设计的准则,本文给出拓展设计Lee偏差的下界.此外,还通过一些数值例子来进一步说明、解释本文的理论结果.     

倍扩设计的构造及其均匀性

均匀设计作为一种空间填充设计,由于具有灵活的试验次数和模型稳健性被广泛运用到各个领域.倍扩方法在构造具有优良性质的二水平部分因析设计中起着非常重要的作用.本文将二水平设计的倍扩构造方法推广至四水平,二、四混水平设计,分别提出了四水平和二、四混水平倍扩设计的新概念,在可卷L_2-偏差意义下研究了二水平,四水平,二、四混水平倍扩设计与其初始设计均匀性之间的关系.同时获得这些倍扩设计的可卷L_2-偏差的新下界,这些下界为评价倍扩设计的均匀性提供一个基准.最后讨论了倍扩设计的均匀性.     

Uniform supersaturated design and its construction

Supersaturated designs are factorial designs in which the number of main effects is greater than the number of experimental runs. In this paper, a discrete discrepancy is proposed as a measure of uniformity for supersaturated designs, and a lower bound of this discrepancy is obtained as a benchmark of design uniformity. A construction method for uniform supersaturated designs via resolvable balanced incomplete block designs is also presented along with the investigation of properties of the resulting designs. The construction method shows a strong link between these two different kinds of designs     

Some results on resolvable incomplete block designs

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.     

Constructions of uniform designs by using resolvable packings and coverings

Uniform designs have been widely used in computer experiments, as well as in industrial experiments when the underlying model is unknown. Based on the discrete discrepancy, the link between uniform designs, and resolvable packings and coverings in combinatorial theory is developed. Through resolvable packings and coverings without identical parallel classes, many infinite classes of new uniform designs are then produced.     

Construction of uniform designs without replications

A uniform design scatters its design points evenly on the experimental domain according to some discrepancy measure. In this paper all the design points of a full factorial design can be split into two subdesigns. One is called the complementary design of the other. The complementary design theories of characterizing one design through the other under the four commonly used discrepancy measures are investigated. Based on these complementary design theories, some general rules for searching uniform designs through their complementary designs are proposed. An efficient method to check if a design has repeated points is introduced and a modified threshold-accepting algorithm is proposed to search uniform or nearly uniform designs without replications. The new algorithm is shown to be more efficient by comparing with other existing methods. Many new uniform or nearly uniform designs without replications are tabulated and compared.     

Some properties of double designs in terms of lee discrepancy

Doubling is a simple but powerful method of constructing two-level fractional factorial designs with high resolution. This article studies uniformity in terms of Lee discrepancy of double designs. We give some linkages between the uniformity of double design and the aberration case of the original one under different criteria. Furthermore, some analytic linkages between the generalized wordlength pattern of double design and that of the original one are firstly provided here, which extend the existing findings. The lower bound of Lee discrepancy for double designs is also given.     

A Construction for Resolvable Designs and Its Generalizations

 In [14], D.K. Ray-Chaudhuri and R.M. Wilson developed a construction for resolvable designs, making use of free difference families in finite fields, to prove the asymptotic existence of resolvable designs with index unity. In this paper, generalizations of this construction are discussed. First, these generalizations, some of which require free difference families over rings in which there are some units such that their differences are still units, are used to construct frames, resolvable designs and resolvable (modified) group divisible designs with index not less than one. Secondly, this construction method is applied to resolvable perfect Mendelsohn designs. Thirdly, cardinalities of such sets of units are investigated. Finally, composition theorems for free difference families via difference matrices are described. They can be utilized to produce some new examples of resolvable designs.     

A note on optimal foldover four-level factorials

The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L ₂-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L ₂-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient algorithms for searching the optimal four-level foldover plans.     

Wrap-Around L2-Discrepancy of Random Sampling, Latin Hypercube and Uniform Designs

For comparing random designs and Latin hypercube designs, this paper con- siders a wrap-around version of the L₂-discrepancy (WD). The theoretical expectation and variance of this discrepancy are derived for these two designs. The expectation and variance of Latin hypercube designs are significantly lower than those of the corresponding random designs. We also study construction of the uniform design under the WD and show that one-dimensional uniform design under this discrepancy can be any set of equidistant points. For high dimensional uniform designs we apply the threshold accepting heuristic for finding low discrepancy designs. We also show that the conjecture proposed by K. T. Fang, D. K. J. Lin, P. Winker, and Y. Zhang (2000, Technometrics) is true under the WD when the design is complete.     

Construction of Sudoku designs and Sudoku-based uniform designs

In this paper, an easy and effective construction method of Sudoku designs with any order is provided based on the right shift operator. Based on the constructed Sudoku designs, a class of Sudoku-based uniform designs is constructed. Moreover, the properties of the constructed Sudoku designs and Sudoku-based uniform designs are investigated, it is shown that both the constructed Sudoku designs and Sudoku-based uniform designs are uniform designs in terms of discrete discrepancy.