K-circulant Supersaturated Designs Research Articles

Multi-level and mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) constitute an important class of fractional factorial designs that could be extremely useful in factor screening experiments. Most of the existing studies have focused on balanced designs. This paper provides a new lower bound for the \(E(f_{NOD})\)-optimality measure of SSDs with general run sizes. This bound is a generalization of existing bounds since it is applicable to both balanced and unbalanced designs. Optimal multi and mixed-level, balanced and nearly balanced SSDs are constructed by applying a k-circulant type methodology. Necessary and sufficient conditions are introduced for the generator vectors, in order to pre-ensure the optimality of the constructed k-circulant SSDs. The provided lower bounds were used to measure the efficiency of the generated designs. The presented methodology leads to a number of new families of improved SSDs, providing tools for directly constructing optimal or nearly-optimal k-circulant designs by just checking the corresponding generator vector.

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.

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.

Constructing [formula omitted]-optimal and minimax-optimal k-circulant supersaturated designs via multi-objective tabu search

A multi-objective optimization procedure based on tabu search method was developed for constructing E(s2)-optimal and minimax-optimal k-circulant supersaturated designs. The construction method is based on cyclic generators. Comparisons with other existing designs are made, demonstrating the effectiveness of our method.

Construction of Efficient Mixed-Levelk-Circulant Supersaturated Designs

Abstract An algorithm to construct efficient balanced mixed-level k-circulant supersaturated designs with m factors and n runs is presented in this article. The algorithm generates efficient mixed-level k-circulant supersaturated designs very fast. Using the proposed algorithm many mixed-level, balanced supersaturated designs are constructed and catalogued. A list of many optimal and near optimal, mixed-level supersaturated designs is also provided for m ≤ 60.

QUASI-CYCLIC CODES FROM CYCLIC-STRUCTURED DESIGNS WITH GOOD PROPERTIES

In this paper, one-generator binary quasi-cyclic (QC) codes are explored by statistical tools derived from design of experiments. A connection between a structured cyclic class of statistical designs, k-circulant supersaturated designs and QC codes is given. The mathematical structure of the later codes is explored and a link between complementary dual binary QC codes and E(s2)-optimal k-circulant supersaturated designs is established. Moreover, binary QC codes of rate 1/3, 1/4, 1/5, 1/6 and 1/7 are found by utilizing a genetic algorithm. Our approach is based on a search for good or best codes that attain the current best-known lower bounds on the minimum distance of linear codes, formulated as a combinatorial optimization problem. Surveying previous results, it is shown, that our codes reach the current best-known lower bounds on the minimum distance of linear codes with the same parameters.

Optimal k-circulant supersaturated designs

Optimal k-circulant supersaturated designs have been constructed in literature using computer intensive methods. A systematic method of construction for multi-level experiments based on balanced incomplete block designs is presented in this paper. The method is also applicable to two-level experiments. Illustrative examples are also 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.

Exploring [formula omitted]-circulant supersaturated designs via genetic algorithms

E ( s 2 ) -optimal or near optimal, two level k-circulant supersaturated designs are explored by means of genetic algorithms. All known k-circulant classes for k = 2 , … , 6 have been rebuilt and improved. The successful application of genetic algorithms is further illustrated by the construction of several k-circulant supersaturated designs for k = 7 , 8 .

Construction and analysis of [formula omitted] efficient supersaturated designs

In this paper, we construct supersaturated designs for large numbers of two-level factors and 10 ⩽ n ⩽ 22 runs by augmenting k-circulant designs [Liu, Y., Dean, A.M., 2004. k-circulant supersaturated designs. Technometrics 46, 32–43] with interaction columns or by deleting columns from k-circulant designs. Most of the designs presented have Es 2 efficiencies above 0.90 and they extend the range of efficient supersaturated designs available in the literature. Difficulties encountered in the use of supersaturated designs in detecting active factors are addressed. We show that, when only one factor is active, the regression technique of forward selection is guaranteed to select the correct factor as active under the idealized conditions that non-active factors have negligible effects and the errors are small. Under similar conditions, we derive bounds on the maximum allowable correlation between the columns of the model matrix that guarantee the correct selection of the “most active” factor when two or more factors are non-negligible. Further, we obtain conditions for the correct selection of the two most active factors using subset selection in regression. A number of designs that satisfy these conditions are identified.

K-Circulant Supersaturated Designs

A class of supersaturated designs called k-circulant designs is explored. These designs are constructed from cyclic generators by cycling k elements at a time. The class of designs includes many Es2-optimal designs, some of which are already known and some of which are more efficient than known designs for model estimation under factor sparsity. Generators for the most efficient designs are listed, and projection properties of some of the designs are explored. We also illustrate that some k-circulant supersaturated designs can be augmented with interaction columns to produce efficient designs for a larger number of factors or for estimating interactions.