Minimum Aberration Research Articles

Minimum G Aberration Design Construction and Design Tables for 24 Runs

Tang and Deng (1999) proposed minimum G aberration as a generalized version of the usual minimum aberration criterion by Fries and Hunter (1980), thereby providing a criterion that can be applied to both regular and nonregular fractional factorial designs. This paper presents the excursion-at-target algorithm for the construction of minimum G aberration designs. A new criterion is introduced that is computationally simpler to apply than minimum G aberration, and it is put to use in the excursion-at-target algorithm as a time-saving surrogate. Although the algorithm cannot guarantee minimum G aberration, we have shown that excursion-at-target performs well and often results in the best designs. A complete table of minimum or near-minimum G aberration designs of 24 runs accommodating between three and 23 factors is given. Minimum or near-minimum G2 and G4 aberration designs are also tabulated.

Characterization of minimum aberration mixed factorials in terms of consulting designs

In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain combinatorial identities that relate the wordlength pattern of a regular mixed factorial design to that of its consulting design. According to these identities, we further-more establish the general and unified rules for identifying minimum aberration mixed factorial designs through their consulting designs. It is an improvement and generalization of the results in Mukerjee and Wu (2001).

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2 k − p III initial designs with their complementary designs when Open image in new window , where n=2 k − p . By applying these results, we are able to characterize 2 k − p III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2 k − p III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when Open image in new window and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration.

Constructing non-regular robust parameter designs

In recent years there has been considerable attention paid to robust parameter design as a strategy for variance reduction. Of particular concern is the selection of a good experimental plan in light of the two different types of factors in the experiment (control and noise) and the asymmetric manner in which effects of the same order are treated. Recent work has focussed on the selection of regular fractional factorial designs in this setting. In this article, we consider the construction and selection of optimal non-regular experiment plans for robust parameter design. Our approach defines the word-length pattern for non-regular fractional factorial designs with two different types of factors which allows for the choice of optimal design to emphasize the estimation of the effects of interest. We use this new word-length pattern to rank non-regular robust parameter designs. We show that one can easily find minimum aberration robust parameter designs from existing orthogonal arrays. The methodology is demonstrated by finding optimal assignments for control and noise factors for 12, 16 and 20-run orthogonal arrays.

Uniformity pattern and related criteria for two-level factorials

In this paper, the study of projection properties of two-level factorials in view of geometry is reported. The concept of uniformity pattern is defined. Based on this new concept, criteria of uniformity resolution and minimum projection uniformity are proposed for comparing two-level factorials. Relationship between minimum projection uniformity and other criteria such as minimum aberration, generalized minimum aberration and orthogonality is made explict. This close relationship raises the hope of improving the connection between uniform design theory and factorial design theory. Our results provide a justification of orthogonality, minimum aberration, and generalized minimum aberration from a natural geometrical interpretation.

Weak minimum aberration and maximum number of clear two-factor interactions in 2IVm-p designs

Both the clear effects and minimum aberration criteria are the important rules for the design selection. In this paper, it is proved that some 2 IV m-p designs have weak minimum aberration, by considering the number of clear two-factor interactions in the designs. And some conditions are provided, under which a 2 IV m-p design can have the maximum number of clear two-factor interactions and weak minimum aberration at the same time. Some weak minimum aberration 2 IV m-p designs are provided for illustrations and two nonisomorphic weak minimum aberration 2 IV 13-6 designs are constructed at the end of this paper.

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.

Another Look at Projection Properties of Hadamard Matrices

Suppose a large number of factors (q) is examined in an experimental situation. It is often anticipated that only a few (k) of these will be important. Usually, it is not known which of the q factors will be the important ones, that is, it is not known which k columns of the experimental design will be of further interest. Screening designs are useful for such situations. It is of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties, since it helps experimenters in selecting a screening design with favorable properties. In this paper we study all the classes of inequivalent projections of certain two-level orthogonal arrays that arise from Hadamard matrices, using well known statistical criteria, such as generalized resolution and generalized minimum aberration. We also pay attention to each projection's distinct runs. Results are given for orthogonal arrays with n = 16, 20 and 24 runs, when they are projected into small dimensions. Useful remarks on design selection are made based on the frequency of appearance of each projection design in every Hadamard matrix.

Minimum aberration blocking of regular mixed factorial designs

It is known by Zhang and Park (J. Statist. Plann. Inference 91 (2000) 107) that there are no minimum aberration (MA) designs with respect to both treatments and blocks for blocked regular mixed-level factorial designs. So it should be compromised between the block wordlength pattern and treatment wordlength pattern. Two methods are considered in this article. The first is MA blocking scheme of an MA design. The other is to combine the components of the two wordlength pattern vectors into one combined wordlength pattern according to the modified hierarchical assumptions and an appropriate ordering of the numbers of alias or confounding relations. The relationship between the two types of optimal blocked designs is investigated. A complete catalogue of optimal blocked regular mixed factorial designs of the above two types with 16 or 32 runs is given.

Construction of Two-Level Split-Lot Fractional Factorial Designs for Multistage Processes

Split-lot designs, also known as multiway split-unit designs, are useful in factorial experiments where there are multiple processing stages. Such experiments occur, for example, in the fabrication of integrated circuits in the semiconductor industry and in product assembly. The designs have a split-plot structure at each stage so that settings of the factors are applied to sublots of experimental runs rather than to individual runs. This allows experimental runs to be processed at much less expense than if the runs were completely randomized. In this article, split-lot designs are constructed for two, three, and four processing stages. The designs have minimum aberration under the split-lot structure and minimize in various senses the confounding of main effects and two-factor interactions with the sublots at each stage.

Nonregular designs from Hadamard matrices and their estimation capacity

Deng and Tang (1999) proposed the generalized minimum aberration (GMA) criterion to assess fractional factorial designs, and a design with GMA is often regarded as the best. However, there exist situations where some other designs may suit practical needs better. In this article, we propose an algorithm to sequentially examine designs obtained from Hadamard matrices under estimation capacity (EC) and provide designs with maximum or high EC for various combinations of run-size and number-of-factors. The usefulness of maximum or high EC designs is discussed.

Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs

Recently, Xu and Wu (2001) presented generalized minimum aberration criterion for comparing and selecting general fractional factorial designs. This criterion is defined using a set of χ u (D) values, called J-characteristics by us. In this paper, we find a set of linear equations that relate the set of design points to that of J-characteristics, which implies that a factorial design is uniquely determined by its J-characteristics once the orthonormal contrasts are designated. Thereto, a projection justification of generalized minimum aberration is established. All of these conclusions generalize the results for two-level symmetrical factorial designs in Tang (2001).

Geometric isomorphism and minimum aberration for factorial designs with quantitative factors

Factorial designs have broad applications in agricultural, engineering and scientific studies. In constructing and studying properties of factorial designs, traditional design theory treats all factors as nominal. However, this is not appropriate for experiments that involve quantitative factors. For designs with quantitative factors, level permutation of one or more factors in a design matrix could result in different geometric structures, and, thus, different design properties. In this paper indicator functions are introduced to represent factorial designs. A polynomial form of indicator functions is used to characterize the geometric structure of those designs. Geometric isomorphism is defined for classifying designs with quantitative factors. Based on indicator functions, a new aberration criteria is proposed and some minimum aberration designs are presented.

Sn− m Designs containing clear main effects or clear two-factor interactions

For a fixed number of runs, when can designs have clear main effects or clear two-factor interactions (in brief, 2fi's)? This paper gives the maximum value of n in s n− m designs containing clear main effects or clear 2fi's, where s is any prime or prime power. It is a generalization of the related results in Chen and Hedayat (J. Statist. Plann. Inference 75 (1998) 147–158) for two-level designs. It is further concluded that the weak minimum aberration designs have a maximum number of clear main effects for two-level designs. A collection of designs containing most clear main effects or clear 2fi's for 16, 32, 27, and 81 runs is given.

Multistratum fractional factorial split-plot designs with minimum aberration and maximum estimation capacity

This paper introduces minimum secondary aberration (MSA) and maximum secondary estimation capacity (MSEC) criteria for discriminating among rival nonisomorphic regular multistratum fractional factorial split-plot (FFSP) designs. Some general rules for identifying MSA or MSEC multistratum FFSP designs through their consulting designs are also established. It is an improvement and generalization of the related results in (Statist. Sinica 12 (2002) 885). The comparison between the MSEC criterion and that of Mukerjee and Fang (2002) is briefly given.

Some theory and the construction of mixed-level supersaturated designs

Supersaturated design is a form of fractional factorial design and has recently received much interest because of its potential in factor screening experiments. This paper mainly concerns the existing criteria for mixed-level designs, i.e. the χ 2( D) criterion (for design D) proposed by Yamada and Matsui (J. Statist. Plann. Inference 104 (2002) 459), the E( f NOD) criterion proposed by Fang et al. (Metrika 58 (2003b) 279) and the minimum moment aberration (MMA) proposed by Xu (Statist. Sinica 13 (2003) 691). A lower bound of χ 2( D) is obtained along with the sufficient and necessary condition for achieving it. The connections between χ 2( D) and other criteria are discussed. Especially, it is shown that χ 2( D) and the second power moment K 2( D) are in fact equivalent to each other and hence the two criteria, χ 2( D) and MMA, share the same optimal supersaturated designs, which can be constructed from saturated orthogonal arrays and other supersaturated designs.

Discrete discrepancy in factorial designs

Discrepancy measure can be utilized as a uniformity measure for comparing factorial designs. A so-called discrete discrepancy has been used to evaluate the uniformity of factorials. In this paper we give linkages among uniformity measured by the discrete discrepancy, generalized minimum aberration, minimum moment aberration and uniformity measured by the centered L2-discrepancy/the wrap-around L2-discrepancy. These close linkages provide a significant justification for the discrete discrepancy used to measure uniformity of factorial designs.

The Design of Blocked Fractional Factorial Split-Plot Experiments

In industrial screening experiments, if some factors are hard to vary and others are easy to vary, randomization restrictions may lead to the use of fractional factorial split-plot (FFSP) designs. Blocked FFSP (BFFSP) designs arise when all runs cannot be performed under homogeneous conditions. This article introduces three approaches for blocking FFSP designs. A catalog of designs is presented in which BFFSP designs are ranked according to the minimum aberration criterion. In addition, additional optimality criteria are used to assist in the ranking procedure. The case study that motivated this research is also discussed in detail.

The design of a triplet with a shape-dependent third-order aberration optimization technique

We have derived a series of shape-dependent third-order aberration equations involving lens thickness. These equations are used in a simple and direct method to correct real aberrations and to find the minimum aberration for triple design that has been proposed. These calculated third-order aberrations can exactly meet the target values for each stage by means of the damped least-squares method. The shape of the three elements permits the control of three third-order aberrations: spherical aberration, coma and astigmatism. The spherical aberration is targeted first. The minimal value of the real on-axis aberration is obtained. Similarly, the coma and astigmatism target values are adjusted so as to force the full field angle real off-axis aberrations down to a minimum. Finally, the manual adjustment of the lens thickness and air spacing is used to attain the minimum aberration at the 0.7 field angle. To verify the method, two triplet design examples are presented.

Minimum aberration construction results for nonregular two-level fractional factorial designs

Nonregular two‐level fractional factorial designs are designs which cannot be specified in terms of a set of defining contrasts. The aliasing properties of nonregular designs can be compared by using a generalisation of the minimum aberration criterion called minimum G2‐aberration. Until now, the only nontrivial designs that are known to have minimum G2‐aberration are designs for n runs and m ≥ n−5 factors. In this paper, a number of construction results are presented which allow minimum G2‐aberration designs to be found for many of the cases with n = 16, 24, 32, 48, 64 and 96 runs and m ≥ n/2−2 factors.