Showing papers on "Orthogonal array published in 1973"

Some Combinatorial Problems of Arrays and Applications to Design of Experiments

Abstract: Publisher Summary This chapter discusses the combinatorial problems of arrays and applications to design of experiments. The combinatorial arrangements are known as hypercube, and more generally as orthogonal arrays. They are applied in the construction of confounded symmetrical and asymmetrical factorial designs, and multifactorial designs. The chapter discusses some arrays with a number of other combinatorial structures slightly weaker than that of orthogonal arrays and explains their use in the design of experiments. It also reviews the use of orthogonal arrays with a variable number of symbols in the rows. The chapter also discusses the concepts of semi-orthogonal arrays, balanced arrays, semi-balanced arrays, and partially balanced arrays.

Balanced Arrays and Orthogonal Arrays

Abstract: We point out the relation between the theory of balanced arrays and various combinatorial areas of design of experiment. Recalling some combinatorial theorems from Srivastava [1971], we apply these to prove a class of new and simple but rather stringent results on the existence of balanced arrays. Applications to the special case of orthogonal arrays are also considered.

Optimal balanced 2 7 fractional factorial designs of resolution V , with N ≦42

Abstract: In this paper, we present a class of fractional factorial designs of the 27 series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).

Optimal balanced 27 fractional factorial designs of resolution V, 49 ≤ N ≤55

Abstract: In this paper, we present a class of fractional factorial designs of the 27 series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).

Some Classical and Modern Topics in Finite Geometrical Structures: A Brief Survey

Abstract: Publisher Summary This chapter discusses the classical and modern topics in finite geometrical structures. It presents a theorem that states the equivalence of finite affine planes and complete sets of mutually orthogonal Latin squares. The classification of projective planes is useful for several reasons: (1) to bring some order to the large number of planes that have already been discovered, (2) to suggest the study of some new characteristic properties of the various types of planes, (3) to enable the discovery of new kinds of planes that would provide examples of planes belonging to some a priori possible class. The properties of the set of orthogonal Latin squares can be translated into properties of the corresponding planes. The notions of a Latin hypercube and of a set of mutually orthogonal Latin hypercubes have also been developed.

On the Problem of Construction and Uniqueness of Saturated Designs

Abstract: Publisher Summary This chapter discusses the problem of construction and uniqueness of saturated designs. In other terminology, these designs are known as orthogonal arrays of strength t = R − 1, in two symbols 2K-p columns, and K rows. The value p corresponds to the number of generators used for the construction. In the usual terminology of orthogonal arrays, p denotes the number of points beyond the points belonging to the identity matrix that are no-t-dependent in the projective space under consideration. The uniqueness and optimal size properties of orthogonal arrays refer only to arrays constructible by geometrical methods. There is no assurance that they hold in general.