Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

Some Properties of β-wordlength Pattern for Four-level Designs

Fractional factorial designs have played a prominent role in the theory and practice of experimental design.For designs with qualitative factors under an ANOVA model,the minimum aberration criterion has been frequently used;however,for designs with quantitative factors,a polynomial regression model is often established,thus theβ-wordlength pattern can be employed to compare different fractional factorial designs.Although theβ-wordlength pattern was introduced in 2004,its properties have not been investigated extensively.In this paper,we will present some properties ofβ-wordlength pattern for four-level designs.These properties can help find better designs with quantitative factors.     

Some results on 4~m2~n designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs,and it has become an active research issue in recent years.Tang et al.derived upper and lower bounds on the maximum number of clear two-factor interactions(2fi's) in 2n-(n-k) fractional factorial designs of resolutions III and IV by constructing a 2n-(n-k) design for given k,which are only restricted for the symmetrical case.This paper proposes and studies the clear effects problem for the asymmetrical case.It improves the construction method of Tang et al.for 2n-(n-k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components(2fic's) in 4m2n designs with resolutions III and IV.The lower bounds are achieved by constructing specific designs.Comparisons show that the number of clear 2fic's in the resulting design attains its maximum number in many cases,which reveals that the construction methods are satisfactory when they are used to construct 4m2n designs under the clear effects criterion.     

Construction of optimal supersaturated designs by the packing method

A supersaturated design is essentially a factorial design with the equal occurrence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently unrelated kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties.     

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.     

CONFOUNDING STRUCTURE OF TWO-LEVEL NONREGULAR FACTORIAL DESIGNS

In design theory,the alias structure of regular fractional factorial designs is elegantly described with group theory.However,this approach cannot be applied to nonregular designs directly. For an arbi...     

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）.     

Minimal regular 2-graphs and applications

A 2-graph is a hypergraph with edge sizes of at most two. A regular 2-graph is said to be minimal if it does not contain a proper regular factor. Let f2(n) be the maximum value of degrees over all minimal regular 2-graphs of n vertices. In this paper, we provide a structure property of minimal regular 2-graphs, and consequently, prove that f2(n) = n 3-i/3, where 1 ≤ i ≤ 6, i ≡ n (mod 6) and n ≥ 7, which solves a conjecture posed by Fan, Liu, Wu and Wong. As applications in graph theory, we are able to characterize unfactorable regular graphs and provide the best possible factor existence theorem on degree conditions. Moreover, fa(n) and the minimal 2-graphs can be used in the universal switch box designs, which originally motivated this study.     

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.     

Some Optimal Block-Factorial Designs

The theory of optimal design plays a fundamental role in experimental design. Some of the results have been widely applied to the realistic world. In[1], Kiefer proved the optimality of some block designs. In this paper some block-factorial designs are discussed. When a block design d_2(v_2,b,k) is superimposed on other block design d_1(v_1,b,k), the resulting structure is called a block-factorial design and is abbreviated as d_1*d_2. The collection of all such designs is denoted by π(v_1,v_2, bk) or Ω. If d_1,d_2 are uniform and they are orthogonal to each other, the structure can be considered as an orthogonal design with three factors. In[2] Cheng pointed out that     

Detailed wordlength pattern of regular fractional factorial split-plot designs in terms of complementary sets

With reference to regular fractional factorial split-plot designs, we consider a detailed wordlength pattern taking due cognizance of the distinction between the whole-plot and sub-plot factors. A generalized version of the MacWilliams’ identity is employed to express the detailed wordlength pattern in terms of complementary sets. Several special features make this result intrinsically different from the corresponding one in classical fractional factorial designs where all factors have the same status. An application to robust parameter designs is indicated and examples given.     

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.     

可卷型L_2-偏差的水平置换方法构造混合水平设计

提出了一种通过置换因子的水平来构造具有较小可卷型L_2-偏差的混合水平均匀设计的新方法.首先建立了混合水平设计的平均可卷型L_2-偏差与广义字长型的定量关系,并以具有较小广义字长型的混合水平设计为初始设计,对其作水平置换,计算其可卷型L_2-偏差,找到具有最小的可卷型L_2-偏差的设计就是相对较好的设计.为了使算法更加有效,还运用了可卷型L_2-偏差的两个性质.数值结果显示通过这种方法构造的设计在可卷型L_2-偏差下表现良好.     

An algorithmic approach to finding factorial designs with generalized minimum aberration

Factorial designs are arguably the most widely used designs in scientific investigations. Generalized minimum aberration (GMA) and uniformity are two important criteria for evaluating both regular and non-regular designs. The generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. Based on an analytical expression between the generalized wordlength pattern and a uniformity measure, this paper converts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also (nearly) optimal under the uniformity measure.     

Theory of optimal blocking for fractional factorial split-plot designs

The issue of optimal blocking for fractional factorial split-plot (FFSP) designs is considered under the two criteria of minimum aberration and maximum estimation capacity. The criteria of minimum secondary aberration (MSA) and maximum secondary estimation capacity (MSEC) are developed for discriminating among rival nonisomorphic blcoked FFSP designs. A general rule for identifying MSA or MSEC blocked FFSP designs through their blocked consulting designs is established.     

A fresh look at effect aliasing and interactions: some new wine in old bottles

Interactions and effect aliasing are among the fundamental concepts in experimental design. In this paper, some new insights and approaches are provided on these subjects. In the literature, the “de-aliasing” of aliased effects is deemed to be impossible. We argue that this “impossibility” can indeed be resolved by employing a new approach which consists of reparametrization of effects and exploitation of effect non-orthogonality. This approach is successfully applied to three classes of designs: regular and nonregular two-level fractional factorial designs, and three-level fractional factorial designs. For reparametrization, the notion of conditional main effects (cme’s) is employed for two-level regular designs, while the linear-quadratic system is used for three-level designs. For nonregular two-level designs, reparametrization is not needed because the partial aliasing of their effects already induces non-orthogonality. The approach can be extended to general observational data by using a new bi-level variable selection technique based on the cme’s. A historical recollection is given on how these ideas were discovered.