Journal of Statistical Theory and Practice

, Volume 1, Issue 3–4, pp 329–346 | Cite as

Two Polynomial Representations of Experimental Design

  • Roberto NotariEmail author
  • Eva Riccomagno
  • Maria-Piera Rogantin


In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the literature: Gröbner bases and indicator functions. We briefly describe them both, how they are used in the analysis and planning of a design and how to switch between them. Examples include fractions of full factorial designs and designs for mixture experiments.

AMS Subject Classification

62K15 13P10 


Algebraic Statistics Factorial design Gröbner basis Indicator function Mixture design 


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  1. Char, B., Geddes, K., Gonnet, G., Leong, B., Monogan, M., Watt, S., 1991. MAPLE V Library Reference Manual. Springer-Verlag, New York.CrossRefGoogle Scholar
  2. Cheng, S.-W., Ye, K. Q., 2004. Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. The Annals of Statistics, 32 (5), 2168–2185.MathSciNetCrossRefGoogle Scholar
  3. CoCoATeam, 2005. CoCoA: a system for doing Computations in Commutative Algebra. Available at Scholar
  4. Cox, D., Little, J., O’Shea, D., 2005. Using algebraic geometry, 2nd Edition. Springer-Verlag, New York.zbMATHGoogle Scholar
  5. Cox, D. A., Little, J. B., O’Shea, D., 1997. Ideals, Varieties, and Algorithms, 2nd Edition. Springer-Verlag, New York.CrossRefGoogle Scholar
  6. Draper, N. R., Pukelsheim, F., 1998. Mixture models based on homogeneous polynomials. J. Statist. Plann. Inference, 71 (1–2), 303–311.MathSciNetCrossRefGoogle Scholar
  7. Fontana, R., Pistone, G., Rogantin, M.-P., 1997. Algebraic analysis and generation of two-levels designs. Statistica Applicata, 9 (1), 15–29.Google Scholar
  8. Fontana, R., Pistone, G., Rogantin, M. P., 2000. Classification of two-level factorial fractions. J. Statist. Plann. Inference 87, (1), 149–172.MathSciNetCrossRefGoogle Scholar
  9. Holliday, T., Pistone, G., Riccomagno, E., Wynn, H. P., 1999. The application of computational algebraic geometry to the analysis of designed experiments: a case study. Comput. Statist., 14 (2), 213–231.MathSciNetCrossRefGoogle Scholar
  10. Kreuzer, M., Robbiano, L., 2000. Computational Commutative Algebra 1, Springer, Berlin-Heidelberg.CrossRefGoogle Scholar
  11. Kreuzer, M., Robbiano, L., 2005. Computational Commutative Algebra 2, Springer, Berlin-Heidelberg.zbMATHGoogle Scholar
  12. Maruri-Aguilar, H., Notari, R., Riccomagno, E., 2007. On the description and identifiability analysis of mixture designs. Statistica Sinica, (1417–1440).Google Scholar
  13. McConkey, B., Mezey, P., Dixon, D., Grenberg, B., 2000. Fractional simplex designs for interaction screening in complex mixtures. Biometrics, 56, 824–832.CrossRefGoogle Scholar
  14. Mora, T., Robbiano, L., 1988. The Gröbner fan of an ideal. Journal of Symbolic Computation, 6, 183–208.MathSciNetCrossRefGoogle Scholar
  15. Pistone, G., Riccomagno, E., Rogantin, M., 2007. In Search for Optimality in Design and Statistics: Algebraic and Dynamical System Methods. Ch. Methods in Algebraic Statistics for the Design of Experiments, pp. 97–132.Google Scholar
  16. Pistone, G., Riccomagno, E., Wynn, H. P., 2001. Algebraic Statistics: Computational Commutative Algebra in Statistics. Chapman&Hall, Boca Raton.zbMATHGoogle Scholar
  17. Pistone, G., Rogantin, M., 2007 a. Algebraic statistics of level codings for fractional factorial designs. J. Statist. Plann. Inference 234–244.Google Scholar
  18. Pistone, G., Rogantin, M., 2007 b. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138 (3), 787–802.MathSciNetCrossRefGoogle Scholar
  19. Pistone, G., Wynn, H. P., 1996. Generalised confounding with Gröbner bases. Biometrika 83 (3), 653–666.MathSciNetCrossRefGoogle Scholar
  20. Scheffé, H., 1958. Experiments with mixtures. J. Roy. Statist. Soc. Ser., B 20, 344–360.MathSciNetzbMATHGoogle Scholar
  21. Scheffé, H., 1963. The simplex-centroid design for experiments with mixtures. J. Roy. Statist. Soc. Ser., B 25, 235–263.MathSciNetzbMATHGoogle Scholar
  22. Tang, B., Deng, L. Y., 1999. Minimum G 2 -aberration for nonregular fractinal factorial designs. The Annals of Statistics, 27 (6), 1914–1926.MathSciNetCrossRefGoogle Scholar
  23. Ye, K. Q., 2003. Indicator function and its application in two-level factorial designs. The Annals of Statistics, 31 (3), 984–994.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2007

Authors and Affiliations

  • Roberto Notari
    • 1
    Email author
  • Eva Riccomagno
    • 2
  • Maria-Piera Rogantin
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoItaly
  2. 2.Dipartimento di MatematicaUniversità di GenovaGenoaItaly

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