Hilbert Bases for Orthogonal Arrays

Abstract

Abstract. In this paper, we relate the problem of generating all 2-level orthogonal arrays of given dimension and force, i.e. elements in OA(n, m), where n is the number of factors and m the force, to the solution of an Integer Programming problem involving rational convex cones. We do not restrict the number of points in the array, i.e. we admit any number of replications. This problem can be theoretically solved by means of Hilbert bases which form a finite generating set for all the elements in in the infinite set OA(n, m). We discuss some examples which are explicitly solved with a software performing Hilbert bases computation.

2-Level Fractional Factorial Designs Which are the Union of Non Trivial Regular Designs

“…If we limit to the different ones we get 192 indicator functions. This number is the same that has been found in Carlini and Pistone (2007), as the total number of orthogonal arrays with 12 runs and strength 2. It is interesting to point out that the understanding of the mechanism underlying the Plackett-Burman designs (m = 5, 12 runs) has allowed to build all the orthogonal arrays of strength 2.…”

“…It follows that the following condition must be satisfied and we limit to the different ones we get 32 design with 11 points and 12 runs, again the same number that has been found in Carlini and Pistone (2007).…”

“…In such a case, we prefer to call F a counting polynomial of the fraction. A systematic algebraic search of orthogonal arrays with replications is discussed in (Carlini and Pistone, 2007). For sake of easy reference in Section 5 below, we quote a couple of specific result about orthogonal arrays.…”

Unions of Orthogonal Arrays and Their Aberrations via Hilbert Bases

“…With respect to the standard notation, we allow the cardinality to vary, because our study will concern the union of two or more OAs, and thus we use the symbol • in place of the cardinality of the fraction. In the case of binary designs, this set has already been considered in [2], where the reader can find also a simple and comprehensive summary of the basic definitions from Combinatorics used here. The generalization to mixed-level designs can be found in [6].…”

Characterizations of indicator functions and contrast representations of fractional factorial designs with multi-level factors

“…However, it is better if we can consider Q, the field of rational numbers, as the coefficients field, because algebraic computations are conducted in Q (or finite fields Z/pZ) for standard computational algebraic software. Another approach is a concept of Hilbert basis presented in [3], where the case of repeated treatments are considered by considering counting functions instead of indicator functions. In this paper, we give generalization of the relations for two-level designs such as (2) and (3) to general multi-level designs for the rational coefficients field Q, and show how to relate the structure of the designs to the structure of their indicator functions.…”