Orthogonal arrays of strength 3 and small run sizes

All mixed (or asymmetric) orthogonal arrays of strength 3 with run size at most 64 are determined.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

On the construction of mixed orthogonal arrays of strength two

The generalized Kronecker sum was used by Wang and Wu (J. Amer. Statist. Assoc. 86 (1991) 450) and Dey and Midha (Statist. Probab. Lett. 28 (1996) 211; Proc. AP Akad. Sci. 5 (2001) 39) to construct mixed orthogonal arrays. We modify their methods to obtain several families of mixed orthogonal arrays. Some new arrays with run size less than 100 are found.     

Asymmetrical Orthogonal Arrays With Run Size 100

Nowadays orthogonal arrays play important roles in statistics and other fields. Usual difference matrices are essential for the construction of many symmetrical or a few asymmetrical orthogonal arrays. But there are also asymmetrical orthogonal arrays which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices were discovered from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented. As an application, a lot of new orthogonal arrays of run size 100 have been constructed.     

On orthogonal arrays attaining Rao's bounds

Rao (1947) provided two inequalities on parameters of an orthogonal array OA(N,m,s,t). An orthogonal array attaining these Rao bounds is said to be complete. Noda (1979) characterized complete orthogonal arrays of t=4 (strength). We here investigate complete orthogonal arrays with s=2 (levels) and general t; and with t=2, 3 and general s.     

Schematic saturated orthogonal arrays obtained by using the contractive replacement method

This paper presents a general method of constructing schematic saturated orthogonal arrays of strength two. We consider a class of mixed saturated orthogonal arrays produced from saturated symmetric orthogonal arrays by using the contractive replacement method, study the Hamming distances of the rows, construct their association schemes, and prove that they are schematic. Some examples are given to illustrate this construction method.     

The Hamming distances of saturated asymmetrical orthogonal arrays with strength 2

Abstract

Orthogonal arrays have many connections to other combinatorial designs and are applied in coding theory, the statistical design of experiments, cryptography, various types of software testing and quality control. In this paper, we present some general methods to find the Hamming distances for saturated asymmetrical orthogonal arrays (SAOAs) with strength 2. As applications of our methods, the Hamming distances of SAOA parents of size less than or equal to 100 are obtained. We also provide the Hamming distances of the SAOAs constructed from difference schemes or by the expansive replacement method. The feasibility of Hamming distances is discussed.     

Some mixed orthogonal arrays obtained by orthogonal projection matrices

The method of orthogonal decomposition of projection matrices is used to construct mixed orthogonal arrays of strength two. Several series of tight orthogonal arrays are constructed by using difference schemes. This method is also used to obtain some new 72-run, 100-run, and 108-run orthogonal arrays.     

Space-filling orthogonal arrays of strength two

This paper considers the use of orthogonal arrays of strength two as experimental designs for fitting a surrogate model. Contrary to standard space-filling designs or Latin hypercube designs, the points of an orthogonal array of strength two are well distributed when they are projected on the two-dimensional faces of the unit cube. The aim is to determine if this property allows one to fit an accurate surrogate model when the computer response is governed by second-order interactions of some input variables. The first part of the paper is devoted to the construction of orthogonal arrays with space-filling properties. In the second part, orthogonal arrays are compared with standard designs for fitting a Gaussian process model.     

A linear programming bound for orthogonal arrays with mixed levels

We show how the Delsarte theory can be used to obtain a linear programming bound for orthogonal arrays with mixed levels. Even for strength 2 this improves on the Rao bound in a large number of cases. The results point to several interesting sets of parameters for which the existence of the arrays is at present undecided.     

Orthogonal arrays of strength three from regular 3-wise balanced designs

The construction given in Kreher, J Combin Des 4 (1996) 67 is extended to obtain new infinite families of orthogonal arrays of strength 3. Regular 3-wise balanced designs play a central role in this construction.     

On the construction of balanced and orthogonal arrays

Two series of three symbol balanced arrays of strength two are constructed. Using special classes of BIB designs, two classes of two symbol orthogonal arrays of strength three are constructed.     

Efficient arrangements of two-level orthogonal arrays in two and four blocks

When orthogonal arrays are used in practical applications, it is often difficult to perform all the designed runs of the experiment under homogeneous conditions. The arrangement of factorial runs into blocks is usually an action taken to overcome such obstacles. However, an arbitrary configuration might lead to spurious analysis results. In this work, the nice properties of two-level orthogonal arrays are taken into consideration and an effective method for arranging experimental runs into two and four blocks of the same size is proposed. This method is based on the so-called J-characteristics of the corresponding array. General theoretical results are given for studying up to four experimental factors in two blocks, as well as for studying up to three experimental factors in four blocks. Finally, we provide best blocking arrangements when the number of the factors of interest is larger, by exploiting the known lists of non-isomorphic orthogonal arrays with two levels and various run sizes.     

Defining equations for two-level factorial designs

Defining equations are introduced in the context of two-level factorial designs and they are shown to provide a concise specification of both regular and nonregular designs. The equations are used to find orthogonal arrays of high strength and some optimal designs. The latter optimal designs are formed in a new way by augmenting notional orthogonal arrays which are allowed to have some runs with a negative number of replicates before augmentation. Defining equations are also shown to be useful when the factorial design is blocked.     

Construction of some new families of nested orthogonal arrays

Nested orthogonal arrays have been used in the design of an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. In this paper, we provide new methods for constructing two types of nested orthogonal arrays.     

An Approach to Constructing Nested Space-Filling Designs for Multi-Fidelity Computer Experiments

Multi-fidelity computer experiments are widely used in many engineering and scientific fields. Nested space-filling designs (NSFDs) are suitable for conducting such experiments. Two classes of NSFDs are currently available. One class is based on special orthogonal arrays of strength two and the other consists of nested Latin hypercube designs. Both of them assume all factors are continuous. We propose an approach to constructing new NSFDs based on powerful (t, s)-sequences. The method is simple, easy to implement, and quite general. For continuous factors, this approach produces NSFDs with better space-filling properties than existing ones. Unlike the previous methods, this method can also construct NSFDs for categorical and mixed factors. Some illustrative examples are given. Other applications of the constructed designs are briefly discussed.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

A Lower Bound to the Measure of Optimality for Main Effect Plans in the Symmetric Factorial Experiments

A design d is called D-optimal if it maximizes det(M _d ), and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this article, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s-level symmetric orthogonal array of strength at least one. Non isomorphic two level MS-optimal orthogonal arrays of strength one with N = 10, 14, and 18 runs, non isomorphic three level MS-optimal orthogonal arrays of strength one with N = 6, 12, and 15 runs and non isomorphic four level MS-optimal orthogonal arrays of strength one with N = 12 runs are also presented.     

Using orthogonal array for constructing three-level search designs

We consider the problem of constructing search designs for 3^m factorial designs. By using projection properties of some three-level orthogonal arrays, some search designs are obtained for 3 ? m ? 11. The new obtained orthogonal search designs are capable of searching and identifying up to four two-factor interactions and estimating them along with the general mean and main effects. The resulted designs have very high searching probabilities; it means that besides the well-known orthogonal structure, they have high ability in searching the true effects.     

Classification of affine resolvable 2-(27, 9, 4) designs

All affine resolvable designs with parameters of the design of the hyperplanes in ternary affine 3-space are enumerated. This enumeration implies the classification (up to equivalence), of all optimal equidistant ternary codes of length 13 and distance 9, as well as all complete orthogonal arrays of strength 2 with 3 symbols, 13 constraints and index 3. Up to isomorphism, there are exactly 68 such designs. The automorphism groups and the rank of the incidence matrices over GF(3) are computed. There are six designs with point-transitive automorphism groups, and one design with trivial group. The affine geometry design is the unique design with lowest 3-rank, and the only design with 2-transitive automorphism group.