Abstract
In this paper, we relate the problem of generating all 2-level orthogonal arrays of given dimension and strength, i.e. elements in OA (N,2n,m), where N is the number of rows, n is the number of factors, and m the strength, to the solution of an integer programming problem involving rational convex cones. We admit any number of replications in the arrays, i.e. we consider not only set but also multi-set. This problem can be theoretically solved by means of Hilbert bases which form a finite generating set for all the elements in the infinite set OA (N,2n,m), N ∈ ℕ. We discuss some examples which are explicitly solved with a software performing Hilbert bases computation.
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References
4ti2 team, 2007. 4ti2 — a software package for algebraic, geometric and combinatorial problems on linear spaces, Available at www.4ti2.de, v. 1.3.1.
Ahmed, Maya and De Loera, Jesús and Hemmecke, Raymond, 2003. Polyhedral cones of magic cubes and squares, Discrete and computational geometry, Springer, Berlin, 25, 25–41.
CoCoATeam, 2007. CoCoA: a system for doing Computations in Commutative Algebra, Available at https://doi.org/cocoa.dima.unige.it, v. 4.7.
Eisenbud D. and Harris J., 1992. Schemes. The language of modern algebraic geometry, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
Fontana R., Pistone G. and Rogantin M.P., 1997. Algebraic analysis and generation of two-levels designs, Stat-istica Applicata, 9, 15–29.
Fontana R., Pistone G. and Rogantin M.P., 2000. Classification of two-level factorial fractions, Journal of Statistical Planning and Inference, 87, 149–172.
Hedayat A.S., Sloane N.J.A. and Stufken J., 1999. Orthogonal arrays. Theory and applications, Springer Series in Statistics, Springer-Verlag, New York.
Hemmecke R. and Weismantel R., 2004. On the computation of Hilbert bases and extreme rays of cones, Tech. Rep. math.CO/0410225, ArXiv.
Mukerjee R. and Wu C.F.J., 2006. A modern theory of factorial designs, Springer Series in Statistics, Springer, New York.
Pistone G., Riccomagno E. and Wynn H.P., 2001. Algebraic Statistics: Computational Commutative Algebra in Statistics, Chapman&Hall.
Pistone G. and Rogantin M., 2007. Indicator function and complex coding for mixed fractional factorial designs, Journal of Statistical Planning and Inference, Accepted for publication.
Pistone G. and Wynn H.P., 1996. Generalised confounding with Gröbner bases, Biometrika, 83, 653–666.
Schrijver A., 1986. Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons Ltd., Chichester, a Wiley-Interscience Publication.
Ye K.Q., 2003. Indicator function and its application in two-level factorial designs, The Annals of Statistics, 31, 984–994.
Ye K.Q., 2004. A note on regular fractional factorial designs, Statistica Sinica, 14, 1069–1074.
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Carlini, E., Pistone, G. Hilbert Bases for Orthogonal Arrays. J Stat Theory Pract 1, 299–309 (2007). https://doi.org/10.1080/15598608.2007.10411842
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DOI: https://doi.org/10.1080/15598608.2007.10411842