Multi-level Supersaturated Designs Research Articles

Construction of Efficient Multi-Level k-Circulant Supersaturated Designs

This article proposes an algorithm to construct efficient balanced multi-level k-circulant supersaturated designs with m factors and n runs. The algorithm generates efficient balanced multi-level k-circulant supersaturated designs very fast. Using the proposed algorithm many balanced multi-level supersaturated designs are constructed and cataloged. A list of many optimal and near optimal, multi-level supersaturated designs is also provided for m ≤ 60 and number of levels (q) ≤10. The algorithm can be used to generate two-level k-circulant supersaturated designs also and some large optimal two-level supersaturated designs are presented. An upper bound to the number of factors in a balanced multi-level supersaturated design such that no two columns are fully aliased is also provided.

Optimal multi-level supersaturated designs through integer programming

This article proposes a novel approach to construct optimal balanced multi-level supersaturated designs using linear integer programming. We illustrate the approach with an example. Using the proposed approach, we construct and catalogue optimal balanced multi-level supersaturated designs in the parametric range m≤100,n≤20 and q≤10.

A general construction of E(f NOD) -optimal multi-level supersaturated designs

The present paper deals with E ( f NOD ) - optimal multi-level supersaturated designs. We present a new technique for the construction of supplementary difference sets. Based on the new supplementary difference sets, we also provide E ( f NOD ) - optimal multi-level supersaturated designs with a large number of columns when compared with other designs. Moreover, these designs retain the equal occurrence property.

Computer Aided Construction of Efficient Multi-Level Supersaturated Designs

Abstract Motivated by the computer search algorithms for constructing two-level supersaturated designs by Heavlin and Finnegan (1993), Li and Wu (1997), Nguyen (1996), Lejeune (2003) and Gupta, Parsad, Kole and Bhar (2008), this paper develops an algorithm to generate multi-level supersaturated designs. Popular E(fnod) and E(λ2) criterion have been used as a measure of non-orthogonality for the designs generated. The algorithm also ensures that no two columns in the designs generated are fully aliased. A catalogue of 120 optimal supersaturated designs for different number of factors m, design runs n, with 5 ≤ n < 16 runs, and different number of factor levels q, with 3 < q < 6, has been prepared. All the designs generated are fnod-optimal; some designs are λ 2-optimal too.

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.

Generalized E(s 2) Criterion for Multilevel Supersaturated Designs

Several extensions of the popular E(s 2) criterion of Booth and Cox (1962) to multilevel supersaturated designs have been advanced in literature. These extensions are not unique due to different ways they measure overall nonorthogonality between all pairs of the columns of the model matrix. We exploit the connection of the E(s 2) criterion with A- and D-optimality that naturally lends itself to a generalized criterion for the multilevel situation in a unified way. The extensions provided in literature follow as special cases of the generalized criterion. A lower bound to the generalized criterion is derived for a wide class of designs, and a method of construction for the symmetrical case is discussed.

Probability of correct model identification in supersaturated design

A criterion for comparing two competitive designs is proposed for symmetric factorial experiment based on the probability of correct identification of active factors. An algorithm for constructing multi-level supersaturated designs which maximize the probability of correct model identification is presented.

Construction of multi-level supersaturated design via Kronecker product

Supersaturated design is a kind of fractional factorial design in which the number of runs is not enough to estimate all the main effects. It has recently received much interest because of its potential in factor screening experiments. In this paper, we propose a method for constructing multi-level supersaturated design via Kronecker product. The resulting design has the good properties such as: its χ 2 value can be computed exactly and the upper bound on the dependency among all design columns is under control. Illustrating examples followed by some detailed discussions are presented.

Multi-level k-circulant Supersaturated Designs

In this paper we present a new method for constructing multi-level supersaturated designs with n rows, m columns and the equal occurrence property. We investigate the existence of multi-level supersaturated designs using a single generator vector and its k-cyclic permutations as rows. We find the conditions needed, in order this vector to generate a balanced supersaturated design. These conditions are simplified for the case of three level supersaturated designs. By using the proposed method three level supersaturated designs are constructed and explored. Moreover, many new, optimal and near optimal, multi-level supersaturated designs are provided as well.

Construction of optimal multi-level supersaturated designs

Author(s): Xu, Hongquan; Wu, CFJ | Abstract: A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu (2001). Optimal supersaturated designs are shown to have a periodic property and general methods for constructing optimal multilevel supersaturated designs are proposed. Inspired by the Addelman-Kempthorne construction of orthogonal arrays, optimal multi-level supersaturated designs are given in an explicit form: columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties.

Large supersaturated designs over prime fields

In the recent literature, multi-level supersaturated designs (SSDs) have been extensively developed and studied. Two construction methods for optimal multi-level SSDs are used and the new 5-level and 7-level designs arising from these methods are compared using several criteria.

Tabu Search and Perturbation Methods in the Construction of Supersaturated Designs

SYNOPTIC ABSTRACTWe adapt the columnwise-pairwise algorithm (CP) of Li and Wu (1997) to the construction of multi- and mixed-level supersaturated designs. One inherent difficulty associated with the use of many heuristic algorithms is premature convergence to a local optima. Meyer and Nachtsheim (1988) discussed the use of simulated annealing method for constructing D-optimal designs. As an extension to their work, this research proposes two modified CP algorithms: tabu-enhanced CP and CP with perturbation. The performance of the two methods is evaluated in the construction of multi-level supersaturated designs. Using these algorithms yields a class of supersaturated designs that is more efficient than existing designs in terms of the optimality criteria. Results are also compared with Fang, Lin, and Ma (2000). The application of the CP algorithms to mixed-level supersaturated designs is examined as well.

On multi-level supersaturated designs

In this paper we present a method of construction E ( f NOD ) -optimal multi-level supersaturated design with n rows, m columns and the equal occurrence property, from a resolvable balanced incomplete block design. A connection between orthogonal arrays and resolvable balanced incomplete block designs is discussed and some E ( f NOD ) -optimal multi-level supersaturated designs are provided.

Optimal multi-level supersaturated designs constructed from linear and quadratic functions

In this paper, we present a method of construction E ( f NOD ) -optimal multi-level supersaturated design with n rows, m columns and the equal occurrence property, using linear and quadratic functions. Using this method, as well as permutations and juxtapositions of the columns, many new E ( f NOD ) -optimal multi-level supersaturated designs are provided.

A new method of construction of multi-level supersaturated designs

Fang et al. (J. Statist. Plann. Inference 86 (2000b) 239) have given a method of construction of multi-level supersaturated designs. Our paper presents a new method of construction of multi-level supersaturated designs, based on Galois field theory. The supersaturated designs constructed in the paper are E( f NOD)-optimal.

A systematical procedure in the construction of multi-level supersaturated design

In the last 2 years, some research results on the construction of multi-level supersaturated designs appeared in the literature (J. Statist. Plann. Inference 81 (1999) 183; 86 (2000) 239; Chinese Ann. Math. B 22 (2001) 183). In this paper we present a systematical procedure in the construction of multi-level supersaturated design under two criteria— E( d 2) and max( d 2), proposed by Lu and Sun (Chinese Ann. Math. B 22 (2001) 183). The procedure can be used to produce more multi-level supersaturated designs than those presented in the literature. It is based on a basic fact that one can induce an E( d 2)-optimal factorial design (saturated or supersaturated) from a resolvable balanced incomplete block design. The result is an extension of the Nguyen (Technometrics 38 (1996) 69) discovery for two-level supersaturated design. A number of resolvable balanced incomplete block designs with corresponding supersaturated (or saturated) factorial designs are presented. Using these factorial designs as bases, more and larger supersaturated designs can be generated in a systematical procedure.

Construction of E (fNOD)-Optimal Supersaturated Designs VIA Room Squares

Supersaturated design is essentially a fractional factorial in which the number of potential effects is greater than the number of runs. And Room square is an important object in combinatorial design theory. We show a link between these two apparently unrelated kinds of designs. E ( fNOD) criterion for comparing supersaturated designs is proposed and a lower bound of E ( fNOD) is obtained as a benchmark of design optimality. It is shown that the E ( fNOD) criterion is an extension of the popular E( s2) and ave x2 criterion (for two- and three-level supersaturated designs respectively). A new construction method for multi-level supersaturated designs via Room squares is also proposed and some properties of the resulting designs are investigated.

SUPERSATURATED DESIGN WITH MORE THAN TWO LEVELS

Supersaturated designs are useful in screening experiments. This paper discusses the topic or multi-level supersaturated design. Two quantities, E(d2) and Df, are proposed to evaluate the optimality of supersaturated designs. A lower bound of E(d2) is obtained with a necessary condition for achieving it. Some E(d2)-optimal supersaturated designs of 3, 4, and 5 levels are given.

On the construction of multi-level supersaturated designs

New criteria of comparing multi-level supersaturated designs are proposed and their properties are studied. A new class of multi-level supersaturated designs are obtained by collapsing a U-type uniform design to an orthogonal array. A global optimization algorithm, the threshold accepting algorithm, is then applied to search for the best supersaturated designs under any prespecified criterion. Examples show that these newly constructed supersaturated designs have good modeling properties.