Boxin Tang

Orthogonal Array-Based Latin Hypercubes

Abstract In this article, we use orthogonal arrays (OAs) to construct Latin hypercubes. Besides preserving the univariate stratification properties of Latin hypercubes, these strength r OA-based Latin hypercubes also stratify each r-dimensional margin. Therefore, such OA-based Latin hypercubes provide more suitable designs for computer experiments and numerical integration than do general Latin hypercubes. We prove that when used for integration, the sampling scheme with OA-based Latin hypercubes offers a substantial improvement over Latin hypercube sampling.

A theorem for selecting oa-based latin hypercubes using a distance criterion

The maximin distance criterion is used for the selection of an OA-based Latin hypercube. For the case in which the underlying orthogonal array is a full factorial design without replication, we construct an OA-based Latin hypercube that reaches the same distance as its parent array.

Complete enumeration of two-level orthogonal arrays of strength d with d + 2 constraints

Enumerating nonisomorphic orthogonal arrays is an important, yet very difficult, problem. Although orthogonal arrays with a specified set of parameters have been enumerated in a number of cases, general results are extremely rare. In this paper, we provide a complete solution to enumerating nonisomorphic two-level orthogonal arrays of strength d with d + 2 constraints for any d and any run size n = λ2 d . Our results not only give the number of nonisomorphic orthogonal arrays for given d and n, but also provide a systematic way of explicitly constructing these arrays. Our approach to the problem is to make use of the recently developed theory of J-characteristics for fractional factorial designs. Besides the general theoretical results, the paper presents some results from applications of the theory to orthogonal arrays of strength two, three and four.

Nonregular Designs With Desirable Projection Properties

In many industrial applications, the experimenter is interested in estimating some of the main effects and two-factor interactions. In this article we rank two-level orthogonal designs based on the number of estimable models containing a subset of main effects and their associated two-factor interactions. By ranking designs in this way, the experimenter can directly assess the usefulness of the experimental plan for the purpose in mind. We apply the new ranking criterion to the class of all non isomorphic two-level orthogonal designs with 16 and 20 runs and introduce a computationally efficient algorithm, based on two theoretical results, which will aid in finding designs with larger run sizes. Catalogs of useful designs with 16, 20, 24, and 28 runs are presented.

Fractional Factorial Designs With Admissible Sets of Clear Two-Factor Interactions

We consider the problem of selecting two-level fractional factorial designs that allow joint estimation of all main effects and some specified two-factor interactions (2fi’s) without aliasing from other 2fi’s. This problem is to find, among all 2 m−p designs with given m and p, those resolution IV designs whose sets of clear 2fi’s contain the specified 2fi’s as subsets. We use a linear graph to represent the set of clear 2fi’s for a resolution IV design, where each line connecting two vertexes represents a clear 2fi between the factors represented by the two vertexes. We call a 2 m−p resolution IV design admissible if its graph is not isomorphic to any proper subgraph of the graph of any other 2 m−p resolution IV design. We show that all even resolution IV designs are inadmissible. We then use a classical subgraph-isomorphism algorithm to determine all admissible designs of 32, 64, and 128 runs. This leads to a concise catalog of all admissible designs of 32 and 64 runs, and a lengthy but substantially reduced list (compared to the number of non-isomorphic designs) for 128 runs. All admissible designs of 128 runs are included in supplementary materials that are available online. A file of R code is available on the first author’s website for generating the actual design tables for any 2 m−p designs of 32, 64, and 128 runs.

Latin Hypercube Designs

Latin hypercubes are a rich class of designs that are suitable for computer experiments and numerical integration. They are easy to generate and achieve maximum stratification in each of the univariate margins of the design region. This article introduces Latin hypercubes, explains how they can be used in computer experiments and numerical integration, and discusses their strengths and weaknesses. A design may not perform well in terms of other criteria such as those of space filling and orthogonality, simply because it is a Latin hypercube. The article concludes with a discussion on some of the methods for constructing Latin hypercubes that have better space-filling or orthogonality properties. Keywords: computer experiment; maximin distance; numerical integration; orthogonal array; space filling

A Method of Constructing Space-Filling Orthogonal Designs

ABSTRACT This article presents a method of constructing a rich class of orthogonal designs that include orthogonal Latin hypercubes as special cases. Two prominent features of the method are its simplicity and generality. In addition to orthogonality, the resulting designs enjoy some attractive space-filling properties, making them very suitable for computer experiments.

Finding MDS-Optimal Supersaturated Designs Using Computer Searches

Supersaturated designs can be evaluated using the minimal dependent sets (MDSs) of columns in the design matrix. This article describes an extensive computer search of balanced two-level supersaturated designs to find those that are MDS-optimal.

A complementary set theory for quaternary code designs

Quaternary code (QC) designs form an attractive class of nonregular factorial fractions. We develop a complementary set theory for characterizing optimal QC designs that are highly fractionated in the sense of accommodating a large number of factors. This is in contrast to existing theoretical results which work only for a relatively small number of factors. While the use of imaginary numbers to represent the Gray map associated with QC designs facilitates the derivation, establishing a link with foldovers of regular fractions helps in presenting our results in a neat form.

A general rotation method for orthogonal Latin hypercubes

&NA; Orthogonal Latin hypercubes provide a class of useful designs for computer experiments. Among the available methods for constructing such designs, the method of rotation is particularly prominent due to its theoretical appeal as well as its space‐filling properties. This paper presents a general method of rotation for constructing orthogonal Latin hypercubes, making the rotation idea applicable to many more situations than the original method allows. In addition to general theoretical results, many new orthogonal Latin hypercubes are obtained and tabulated.