Mixed-level Orthogonal Arrays Research Articles

Design Efficiency of the Asymmetric Minimum Projection Uniform Designs

Highly efficient designs and uniform designs are widely applied in many fields because of their good properties. The purpose of this paper is to study the issue of design efficiency for asymmetric minimum projection uniform designs. Based on the centered L2 discrepancy, the uniformity of the designs with mixed levels is defined, which is used to measure the projection uniformity of the designs. The analytical relationship between the uniformity pattern and the design efficiency is established for mixed-level orthogonal arrays with a strength of two. Moreover, a tight lower bound of the uniformity pattern is presented. The research is relevant in the field of experimental design by providing a theoretical basis for constructing the minimum number of projection uniform designs with a high design efficiency under a certain condition. These conclusions are verified by some numerical examples, which illustrate the theoretical results obtained in this paper.

On the aberrations of mixed level orthogonal arrays with removed runs

Given an orthogonal array we analyze the aberrations of the sub-fractions which are obtained by the deletion of some of its points. We provide formulae to compute the Generalized Word-Length Pattern of any sub-fraction. In the case of the deletion of one single point, we provide a simple methodology to find which the best sub-fractions are according to the Generalized Minimum Aberration criterion. We also study the effect of the deletion of 1, 2 or 3 points on some examples. The methodology does not put any restriction on the number of levels of each factor. It follows that any mixed level orthogonal array can be considered.

Visualization for Confounding of Interactions in Mixed-level Orthogonal Arrays and Its Application to Response Surface

Visualization for Confounding of Interactions in Mixed-level Orthogonal Arrays and Its Application to Response Surface

ORTHOGONAL ARRAYS CONSTRUCTED BY GENERALIZED KRONECKER PRODUCT

In this paper, we propose a new general approach to construct asymmetrical orthogonal arrays, namely generalized Kronecker product. The operation is not usual Kronecker product in the theory of matrices, but it is interesting since the interaction of two columns of asymmetrical orthogonal arrays can be often written out by the generalized Kronecker product. As an application of the method, some new mixed-level orthogonal arrays of run sizes 72 and 96 are constructed.

Generalized minimum aberration mixed-level orthogonal arrays: A general approach based on sequential integer quadratically constrained quadratic programming

ABSTRACTOrthogonal fractional factorial designs and in particular orthogonal arrays (OAs) are frequently used in many fields of application, including medicine, engineering, and agriculture. In this article, we present a methodology and an algorithm to find an OA, of given size and strength, which satisfies the generalized minimum aberration criterion. The methodology is based on the joint use of polynomial counting functions, complex coding of levels, and algorithms for quadratic optimization and puts no restriction on the number of levels of each factor.

A class of mixed orthogonal arrays obtained from projection matrix inequalities

By using the relationship between orthogonal arrays and decompositions of projection matrices and projection matrix inequalities, we present a method for constructing a class of new orthogonal arrays which have higher percent saturations. As an application of the method, many new mixed-level orthogonal arrays of run sizes 108 and 144 are constructed.

An approach of constructing mixed-level orthogonal arrays of strength ⩾ 3

Orthogonal arrays (OAs), mixed level or fixed level (asymmetric or symmetric), are useful in the design of various experiments. They are also a fundamental tool in the construction of various combinatorial configurations. In this paper, we establish a general "expansive replacement method" for constructing mixedlevel OAs of an arbitrary strength. As a consequence, a positive answer to the question about orthogonal arrays posed by Hedayat, Sloane and Stufken is given. Some series of mixed level OAs of strength ≥ 3 are produced.

Approximation of bounds on mixed-level orthogonal arrays

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Approximation of bounds on mixed-level orthogonal arrays

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Three‐level and mixed‐level orthogonal arrays for lean designs

AbstractOrthogonal arrays (OA's) are widely used in design of experiments. Each OA has a specific number of rows that is fixed by the number of factors in the OA and the number of levels in each factor. In a practical application of an industrial experiment, however, because of various operational constraints it could happen that the number of runs of the experiment cannot be set exactly equal to the number of rows of an OA. In this case, a lean design can be used. A lean design is obtained by removing some specific rows and columns from the extended design matrix formed from an OA, so that the resulting sub‐matrix still allows efficient estimation of the effects of some of the factors. Tables for 2‐level lean designs are already available in the literature. In this paper, the authors will investigate 3‐level lean designs and mixed‐level lean designs, and construct tables for such designs for convenient use. Copyright © 2009 John Wiley & Sons, Ltd.

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.

A Simple Method for Constructing Orthogonal Arrays By the Kronecker Sum

In this article, we propose a new general approach to constructing asymmetrical orthogonal arrays, namely the Kronecker sum. It is interesting since a lot of new mixed-level orthogonal arrays can be obtained by this method.

A Class of Nearly Orthogonal Arrays

Orthogonal arrays with mixed levels are very useful in factorial experiments when factors with different numbers of levels are to be investigated. However, one limitation associated with mixed-level orthogonal arrays is that many of them require a large number of runs. In such cases, nearly orthogonal arrays—which generally use fewer runs by slightly relaxing the requirement of orthogonality—may be useful. We propose a definition of nearly orthogonal arrays of strength m, extending the concept of orthogonal arrays of strength m. We focus on the nearly orthogonal arrays involving 2- and 3-level factors with small run sizes. Some useful arrays are tabulated for practical use. These designs are shown to have nice statistical and combinatorial properties.

Orthogonal arrays of size 108 with six-level columns

A special structure of the two level orthogonal array of order 12 is given. By using this structure and the method of constructing mixed-level orthogonal arrays presented by Zhang, Lu and Pang (1999), a lot of mixed-level orthogonal arrays of run size 108 are obtained. Especially, we obtain an orthogonal array of size 108 with 11 6-level columns.

A note on orthogonal arrays obtained by orthogonal decomposition of projection matrices

A method of constructing mixed-level orthogonal arrays is presented by Zhang et al. (Statist. Sinica 2, 595). It is somewhat difficult to use this method to obtain new orthogonal arrays. This paper illustrates with examples the application of the method. Several classes of mixed-level orthogonal arrays are obtained.

An Algorithm for Constructing Orthogonal and Nearly-Orthogonal Arrays With Mixed Levels and Small Runs

Orthogonal arrays are used widely in manufacturing and high-technology industries for quality and productivity improvement experiments. For reasons of run size economy or flexibility, nearly-orthogonal arrays are also used. The construction of orthogonal or nearly-orthogonal arrays can be quite challenging. Most existing methods are complex and produce limited types of arrays. This article describes a simple and effective algorithm for constructing mixed-level orthogonal and nearly-orthogonal arrays that can construct a variety of small-run designs with good statistical properties efficiently.

A replacement scheme on the construction of orthogonal arrays

A method of constructing orthogonal arrays is presented by Zhang, Lu and Pang in 1999. In this paper, the method is developed by introducing a replacement scheme on the construction of orthogonal arrays, and some new mixed-level orthogonal arrays of run size 36 are constructed.

Orthogonal arrays obtained by generalized Hadamard product

In this paper, by using the generalized Hadamard product and the orthogonal decomposition of projection matrix, we propose a new general approach to construct mixed-level orthogonal arrays. As an application of the method, some new mixed-level orthogonal array's of run sizes 36 and 72 are constructed.

To analyze binary data from mixed-level orthogonal arrays

Orthogonal arrays of mixed levels have been found useful in setting up highly fractional factorial experiments. When the responses from such experiments are binary, logistic regression analysis is used to analyze such data. Usually, the maximum likelihood estimates of parameters under the logistic regression model need to be found by an iterative procedure in order to test the importance of factors. In this article, a simple approximate procedure is proposed and compared to the maximum likelihood method with an example.