An efficient method for constructing uniform designs with large size

Many construction methods for (nearly) uniform designs have been proposed under the centered $L_2$ -discrepancy, and most of them are only suitable for constructing designs with small size. This paper proposes a new method, called mixture method (MM), to construct nearly symmetrical/asymmetrical uniform designs with large number of runs and/or large number of factors. The new method has the “better than given” property, i.e., the resulting design is better than existing designs in the sense of the pre-decided criterion. Moreover, the computational speed of MM is faster than most existing methods.     

混水平列扩充设计的均匀性

在许多工业应用中,通常分步应用跟随设计来考察输入(因子)和产出(响应)间的关系.在许多跟随设计中,在跟随阶段可以加入一些另外的两水平或三水平因子,因为它们在初始阶段可能被忽略但又十分重要.文章在均匀性准则下,提出了中心化L2-偏差意义下的混水平列扩充均匀设计,给出了列扩充设计在中心化L2-偏差下的解析表达式及相应的下界,其可作为搜索均匀设计的基准.进一步,具有一个附加区组因子的列扩充设计是均匀的当且仅当未加区组因子时的列扩充设计是均匀的.     

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

New lower bounds for three- and four-level designs under the centered -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

     

基于计算机试验的均匀设计

本文在计算机试验的基础上，提出了最小相关准则和最小距离离差准则，并将信息论中的Hamming距离和Lee距离引入到计算机试验中，证明了均匀设计在Hamming距离下的最优性和部分好格子点均匀设计在Lee距离下的最优性．基于偏差的考虑，给出了一类新的好格子点均匀设计和一个学习算法，利用这个学习算法，给出了基于Lee距离的最小距离离差准则的均匀设计表的构造方法．通过与已有的好格子点均匀设计和循环拉丁方均匀设计作比较，证明了文中的均匀设计在距离和偏差意义下有更好的均匀性．     

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.     

Constructing uniform designs: A heuristic integer programming method

In this paper, the wrap-around _L2$L_2$-discrepancy (WD) of asymmetrical design is represented as a quadratic form, thus the problem of constructing a uniform design becomes a quadratic integer programming problem. By the theory of optimization, some theoretic properties are obtained. Algorithms for constructing uniform designs are then studied. When the number of runs n$n$ is smaller than the number of all level-combinations m$m$, the construction problem can be transferred to a zero–one quadratic integer programming problem, and an efficient algorithm based on the simulated annealing is proposed. When n≥m$n\ge m$, another algorithm is proposed. Empirical study shows that when n$n$ is large, the proposed algorithms can generate designs with lower WD compared to many existing methods. Moreover, these algorithms are suitable for constructing both symmetrical and asymmetrical designs.     

On theL2-Discrepancy for Anchored Boxes

TheL₂-discrepancy for anchored axis-parallel boxes has been used in several recent computational studies, mostly related to numerical integration, as a measure of the quality of uniform distribution of a given point set. We point out that if the number of points is not large enough in terms of the dimension (e.g., fewer than 10⁴points in dimension 30) then nearly the lowest possibleL₂-discrepancy is attained by a pathological point set, and hence theL₂-discrepancy may not be very relevant for relatively small sets. Recently, Hickernell obtained a formula for the expectedL₂-discrepancy of certain randomized low-discrepancy set constructions introduced by Owen. We note that his formula remains valid also for several modifications of these constructions which admit a very simple and efficient implementation. We also report results of computational experiments with various constructions of low-discrepancy sets. Finally, we present a fairly precise formula for the performance of a recent algorithm due to Heinrich for computing theL₂-discrepancy.     

New Lower Bounds to Wrap-around L2-discrepancy for U-type Designs with Three-level

Acta Mathematicae Applicatae Sinica, English Series - The objective of this paper is to study the issue of uniformity on three-level U-type designs in terms of the wrap-around L2-discrepancy. Based...     

Centered -discrepancy of random sampling and Latin hypercube design, and construction of uniform designs

In this paper properties and construction of designs under a centered version of the -discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance.

     

APPLICATION OF MINIMUM PROJECTION UNIFORMITY CRITERION IN COMPLEMENTARY DESIGNS

In this article, we consider the characterization problem in design theory. The objective is to characterize minimum projection uniformity for two-level designs in terms of their complementary designs. Here, the complementary design means a design in which all the Hamming distances of any two runs are the same, which generalizes the concept of a pair of complementary designs in the literature. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity （MPU） rule to assess and compare two-level factorials.     

一种构造三水平因子超饱和设计的准则和算法

在工业统计试验和其他科学试验中,常遇到因子个数多而所允许的试验次数少的情况,这时要用到超饱和设计,以前的文章仅研究了二水平因子的超饱和设计,本文对于三水平因子的超饱和设计提出了一种基于典则相关意义下的优良性准则和构造算法,并给出了试验次数为９和１８可分别安排到２６和３０个因子的三水平超饱和设计表。     

On the Isomorphism of Fractional Factorial Designs

Two fractional factorial designs are called isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and switching the levels of factors. To identify the isomorphism of two s-factor n-run designs is known to be an NP hard problem, when n and s increase. There is no tractable algorithm for the identification of isomorphic designs. In this paper, we propose a new algorithm based on the centered L₂-discrepancy, a measure of uniformity, for detecting the isomorphism of fractional factorial designs. It is shown that the new algorithm is highly reliable and can significantly reduce the complexity of the computation. Theoretical justification for such an algorithm is also provided. The efficiency of the new algorithm is demonstrated by using several examples that have previously been discussed by many others.     

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     

Optimum mechanism for breaking the confounding effects of mixed-level designs

Fractional factorial designs (FFD’s) are no doubt the most widely used designs in the experimental investigations due to their efficient use of experimental runs to study many factors simultaneously. One consequence of using FFD’s is the aliasing of factorial effects. Follow-up experiments may be needed to break the confounding. A simple strategy is to add a foldover of the initial design, the new fraction is called a foldover design. Combining a foldover design with the original design converts a design of resolution r into a combined design of resolution \(r+1\). In this paper, we take the centered \(L_2\)-discrepancy \(({\mathcal {CD}})\) as the optimality measure to construct the optimal combined design and take asymmetrical factorials with mixed two and three levels, which are most commonly used in practice, as the original designs. New and efficient analytical expressions based on the row distance of the \({\mathcal {CD}}\) for combined designs are obtained. Based on these new formulations, we present new and efficient lower bounds of the \({\mathcal {CD}}\). Using the new formulations and lower bounds as the benchmarks, we may implement a new algorithm for constructing optimal mixed-level combined designs. By this search heuristic, we may obtain mixed-level combined designs with low discrepancy.     

Application of minimum projection uniformity criterion in complementary designs for q-level factorials

We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all the Hamming distances of any two distinct runs are the same, and the uniformity pattern proposed by H. Qin, Z. Wang, and K. Chatterjee [J. Statist. Plann. Inference, 2012, 142: 1170–1177] comes from discrete discrepancy for q-level designs. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity rule to assess and compare q-level factorials.     

低汉明重量序列的分布

数n的汉明重量是指n的二进制字符串表达式中数字1的个数,用Ham(n)来表示.低汉明重量序列在密码系统和编码理论中有非常广泛的应用.本文建立了低汉明重量数的序列表达式,并且利用指数和的上界以及Erdos-Turan不等式证明低汉明重量序列的均匀分布性质,从而确保密码算法的随机性和运算效率.     

Uniformity in double designs

In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L2-discrepancy value of D（X） and generalized wordlength pattern of X, which show the uniformity relationship between D（X） and X. In addition, we also obtain lower bounds of centered L2-discrepancy value of D（X）, which can be used to assess uniformity of D（X）.     

任意凸多面体上布点均匀性的度量

均匀性度量是构作均匀设计的基础,本文从距离概念出发,通过对称的方法,得到一种新的距离函数-势函数,并将势函数作为衡量任意凸多面体上布点均匀性好坏的准则.数值例子和多变量Kendall 协和系数检验表明,当试验区域限制在单位立方体上时,势函数与目前常用的两种偏差-中心化L_2-偏差和可卷L_2.偏差在度量布点均匀性方面结论一致.     

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.     

一类正交空间填充设计的构造

空间填充设计在计算机试验中应用十分广泛,当拟合回归模型时,正交的空间填充设计保证了因子效应估计的独立性.基于广义正交设计,文章给出了构造二阶正交拉丁超立方体设计和列正交设计的方法,新构造的设计不仅满足任意两列之间相互正交,还能保证每一列与任一列元素平方组成的列以及任两列元素相乘组成的列都正交.当某些正交的空间填充设计不存在时,具有较小相关系数的近似正交设计可作为替代设计使用.设计构造的灵活性为计算机试验在实践中的广泛应用提供了必要的支持.