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Journal of Statistical Theory and Practice

, Volume 1, Issue 3–4, pp 299–309 | Cite as

Hilbert Bases for Orthogonal Arrays

  • Enrico Carlini
  • Giovanni Pistone
Article

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.

AMS Subject Classification

62K15 13P10 

Keywords

Orthogonal Array Integer Programming Hilbert Basis 

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References

  1. 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.Google Scholar
  2. 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.MathSciNetCrossRefGoogle Scholar
  3. CoCoATeam, 2007. CoCoA: a system for doing Computations in Commutative Algebra, Available at https://doi.org/cocoa.dima.unige.it, v. 4.7.Google Scholar
  4. 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.zbMATHGoogle Scholar
  5. Fontana R., Pistone G. and Rogantin M.P., 1997. Algebraic analysis and generation of two-levels designs, Stat-istica Applicata, 9, 15–29.Google Scholar
  6. Fontana R., Pistone G. and Rogantin M.P., 2000. Classification of two-level factorial fractions, Journal of Statistical Planning and Inference, 87, 149–172.MathSciNetCrossRefGoogle Scholar
  7. Hedayat A.S., Sloane N.J.A. and Stufken J., 1999. Orthogonal arrays. Theory and applications, Springer Series in Statistics, Springer-Verlag, New York.CrossRefGoogle Scholar
  8. Hemmecke R. and Weismantel R., 2004. On the computation of Hilbert bases and extreme rays of cones, Tech. Rep. math.CO/0410225, ArXiv.Google Scholar
  9. Mukerjee R. and Wu C.F.J., 2006. A modern theory of factorial designs, Springer Series in Statistics, Springer, New York.zbMATHGoogle Scholar
  10. Pistone G., Riccomagno E. and Wynn H.P., 2001. Algebraic Statistics: Computational Commutative Algebra in Statistics, Chapman&Hall.zbMATHGoogle Scholar
  11. 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.Google Scholar
  12. Pistone G. and Wynn H.P., 1996. Generalised confounding with Gröbner bases, Biometrika, 83, 653–666.MathSciNetCrossRefGoogle Scholar
  13. Schrijver A., 1986. Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons Ltd., Chichester, a Wiley-Interscience Publication.zbMATHGoogle Scholar
  14. Ye K.Q., 2003. Indicator function and its application in two-level factorial designs, The Annals of Statistics, 31, 984–994.MathSciNetCrossRefGoogle Scholar
  15. Ye K.Q., 2004. A note on regular fractional factorial designs, Statistica Sinica, 14, 1069–1074.MathSciNetzbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  1. 1.Department of MathematicsPolitecnico di TorinoTurinItaly

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