Optimal Mixed-level Supersaturated Designs Research Articles

Extended mixed-level supersaturated designs

This paper considers the study of the optimality of the extended design generated by adding few runs to an existing E(χ2)-optimal mixed-level supersaturated design. This paper covers the work of Gupta et al. (2010, 2012) on extended two-level and s-level supersaturated designs as two special cases. A lower bound to E(χ2) has been obtained for the proposed designs. A small example is presented here attaining the lower bound.

A Tabu Search Algorithm for the Construction of -Optimal Mixed-Levelk-Circulant Supersaturated Designs

We propose a tabu search algorithm for constructing -optimal mixed-level k-circulant supersaturated designs. The method of constructing designs is based on the cyclic development of a generator, which could be either a row or a column vector. Our algorithm was able to construct 19 new -optimal mixed-level k-circulant supersaturated designs. It also found all -optimal k-circulant designs available in the literature.

On construction of optimal mixed-level supersaturated designs

Supersaturated design (SSD) has received much recent interest because of its potential in factor screening experiments. In this paper, we provide equivalent conditions for two columns to be fully aliased and consequently propose methods for constructing $E(f_{\mathrm{NOD}})$- and $\chi^2$-optimal mixed-level SSDs without fully aliased columns, via equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs have been obtained. Furthermore, it is proved that the nonorthogonality between columns of the resulting design is well controlled by the source designs. A rather complete list of newly generated optimal mixed-level SSDs are tabulated for practical use.

Construction of equidistant and weak equidistant supersaturated designs

Supersaturated designs (SSDs) have been highly valued in recent years for their ability of screening out important factors in the early stages of experiments. Recently, Liu and Lin (in Statist Sinica 19:197–211, 2009) proposed a method to construct optimal mixed-level SSDs from smaller multi-level SSDs and transposed orthogonal arrays (OAs). This paper extends their method to construct more equidistant optimal SSDs by replacing the multi-level SSDs and transposed OAs with mixed-level SSDs and general transposed difference matrices, respectively, and then proposes two practical methods for constructing weak equidistant SSDs based on this extended method. A large number of new optimal SSDs can be constructed from these three methods. Some examples are provided and more new designs are listed in “Appendix” for practical use.

An algorithm for constructing mixed-level [formula omitted]-circulant supersaturated designs

Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Recently, Chen and Liu [Chen, J., Liu, M.Q., 2008a. Optimal mixed-level k -circulant supersaturated designs. J. Statist. Plann. Inference 138, 4151–4157] explored the construction of E ( f NOD ) -optimal mixed-level SSDs using k -cyclic generators. As an extension, this paper further studies how to construct optimal mixed-level k -circulant SSDs under the χ 2 ( D ) criterion. A sufficient condition under which the generator vector produces a χ 2 ( D ) -optimal SSD is obtained, and some other properties of the resulting SSDs are investigated. Note that, the construction of optimal mixed-level k -circulant SSDs under the χ 2 ( D ) criterion is much more complicated than that under E ( f NOD ) . Hence, an algorithm for finding generators of (nearly) χ 2 ( D ) -optimal designs is provided, and many (nearly) χ 2 ( D ) -optimal mixed-level SSDs are then constructed.

Group screening method for the statistical analysis of [formula omitted]-optimal mixed-level supersaturated designs

In this paper, we propose the application of group screening methods for analyzing data using E ( f N O D ) -optimal mixed-level supersaturated designs possessing the equal occurrence property. Supersaturated designs are a large class of factorial designs which can be used for screening out the important factors from a large set of potentially active variables. The huge advantage of these designs is that they reduce the experimental cost drastically, but their critical disadvantage is the high degree of confounding among factorial effects. Based on the idea of the group screening methods, the f factors are sub-divided into g “group-factors”. The “group-factors” are then studied using the penalized likelihood statistical analysis methods at a factorial design with orthogonal or near-orthogonal columns. All factors in groups found to have a large effect are then studied in a second stage of experiments. A comparison of the Type I and Type II error rates of various estimation methods via simulation experiments is performed. The results are presented in tables and discussion follows.

A new look at discrete discrepancy

The objective of this paper is to study the issue of discrete discrepancy [Hickernell, F.J., Liu, M.Q., 2002. Uniform designs limit aliasing. Biometrika 89, 893–904; Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003. Optimal mixed-level supersaturated design. Metrika 58, 279–291; Qin, H., Fang, K.T., 2004. Discrete discrepancy in factorial designs. Metrika 60, 59–72], which has wide application to the field of fractional factorial designs. Here we present an improved lower bound to discrete discrepancy.

An algorithmic approach to constructing mixed-level orthogonal and near-orthogonal arrays

Due to run size constraints, near-orthogonal arrays (near-OAs) and supersaturated designs, a special case of near-OA, are considered good alternatives to OAs. This paper shows (i) a combinatorial relationship between a mixed-level array and a non-resolvable incomplete block design (IBD) with varying replications (and its dual, a resolvable IBD with varying block sizes); (ii) the relationship between the criterion E ( d 2 ) proposed by Lu and Sun [Lu, X., Sun, Y., 2001. Supersaturated designs with more than two levels. Chinese Ann. Math. B 22, 183–194] or E ( f N O D ) proposed by Fang et al. [Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003b. Optimal mixed-level supersaturated design. Metrika 58, 279–291] used in the (near-) OA construction and the ( M , S ) -optimality criterion used in the IBD construction; (iii) the derivation of a tighter bound for E ( d 2 ) ; (iv) how to modify the IBD algorithm of Nguyen [Nguyen, N.-K., 1994. Construction of optimal incomplete block designs by computer. Technometrics 36, 300–307] to obtain efficient (near-) OA algorithms. Some new (near-) OAs are presented and some near-OAs are compared with arrays constructed by other authors. Examples showing the use of the constructed arrays are given.

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only few experiments in the preliminary stages of experimentation. This paper explores how to construct E ( f NOD ) -optimal mixed-level SSDs using k-cyclic generators. The necessary and sufficient conditions for the existence of mixed-level k-circulant SSDs with the equal occurrence property are provided. Properties of the mixed-level k-circulant SSDs are investigated, in particular, the sufficient condition under which the generator vector produces an E ( f NOD ) -optimal SSD is obtained. Moreover, many new E ( f NOD ) -optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean [2004. k -circulant supersaturated designs. Technometrics 46, 32–43] for two-level SSDs and the one due to Georgiou and Koukouvinos [2006. Multi-level k-circulant supersaturated designs. Metrika 64, 209–220] for the multi-level case.

Optimal mixed-level supersaturated design with general number of runs

Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Most studies focused mainly on balanced designs. This paper considers SSDs with general run sizes. The lower bounds of the χ 2 proposed by [Yamada, S., Lin, D.K.J., 1999. Three-level supersaturated designs. Statist. Probab. Lett. 45, 31–39] and E ( f NOD ) proposed by [Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003b. Optimal mixed-level supersaturated design. Metrika 58, 279–291] are obtained along with the sufficient and necessary conditions for achieving them, respectively. Some construction methods for optimal mixed-level SSDs with general run sizes are proposed. Many new designs are also tabulated for practical uses.

Optimal mixed-level supersaturated designs and a new class of combinatorial designs

This paper considers the construction of optimal mixed-level supersaturated designs under the E ( χ 2 ) -optimality criterion. According to a lower bound of the E ( χ 2 ) value, a combinatorial construction method is proposed and illustrated. Meanwhile, a new class of combinatorial designs is put forward. Furthermore, some methods for constructing these combinatorial designs which correspond to E ( χ 2 ) -optimal mixed-level supersaturated designs are provided. All of these results extend the catalogue of E ( χ 2 ) -optimal mixed-level supersaturated designs.

Formula omitted]-optimal mixed-level supersaturated designs

This paper introduces an optimality criterion for mixed-level supersaturated designs, called E ( χ 2 ) , which covers E ( s 2 ) for two-level designs and ave ( χ 2 ) for symmetrical designs as two special cases. Some lower bounds of E ( χ 2 ) are obtained. Furthermore, it is shown that the E ( χ 2 ) criterion coincides with other existing criteria for nonregular mixed-level fractional factorial designs, which include minimum moment aberration, generalized minimum aberration, discrete uniformity and design efficiency. Several classes of E ( χ 2 ) -optimal mixed-level supersaturated designs from saturated orthogonal arrays are given.

Construction of some [formula omitted] optimal mixed-level supersaturated designs

The method of juxtaposition is used in the construction of E ( f NOD ) -optimal mixed level supersaturated designs, with the equal occurrence property. Some new supersaturated designs are obtained by this method.

Some combinatorial constructions for optimal mixed-level supersaturated designs

Supersaturated designs are factorial designs in which the number of factors is greater than the number of runs. In this paper, we present a procedure for the construction of mixed-level supersaturated designs from uniformly resolvable pairwise balanced designs. The designs constructed in this way are E (f NOD)-optimal, with the equal occurrence property, and so can be useful in the researchers, and for some practical uses too.

Optimal mixed-level supersaturated design

A supersaturated design is essentially a fractional factorial in which the number of potential effects is greater than the number of runs. In this paper, E(f NOD ) criterion is employed for comparing supersaturated designs from the viewpoint of orthogonality and uniformity, and a lower bound of E(f NOD ) which can serve as a benchmark of design optimality is obtained. It is shown that the existing E(s 2) and ave χ2 criteria (for two- and three-level supersaturated designs respectively) are in fact special cases of this criterion. Furthermore, a construction method for mixed-level supersaturated designs is proposed and some properties of the resulting designs are investigated.