Abstract
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.
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References
Char, B., Geddes, K., Gonnet, G., Leong, B., Monogan, M., Watt, S., 1991. MAPLE V Library Reference Manual. Springer-Verlag, New York.
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.
CoCoATeam, 2005. CoCoA: a system for doing Computations in Commutative Algebra. Available at https://doi.org/cocoa.dima.unige.it.
Cox, D., Little, J., O’Shea, D., 2005. Using algebraic geometry, 2nd Edition. Springer-Verlag, New York.
Cox, D. A., Little, J. B., O’Shea, D., 1997. Ideals, Varieties, and Algorithms, 2nd Edition. Springer-Verlag, New York.
Draper, N. R., Pukelsheim, F., 1998. Mixture models based on homogeneous polynomials. J. Statist. Plann. Inference, 71 (1–2), 303–311.
Fontana, R., Pistone, G., Rogantin, M.-P., 1997. Algebraic analysis and generation of two-levels designs. Statistica Applicata, 9 (1), 15–29.
Fontana, R., Pistone, G., Rogantin, M. P., 2000. Classification of two-level factorial fractions. J. Statist. Plann. Inference 87, (1), 149–172.
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.
Kreuzer, M., Robbiano, L., 2000. Computational Commutative Algebra 1, Springer, Berlin-Heidelberg.
Kreuzer, M., Robbiano, L., 2005. Computational Commutative Algebra 2, Springer, Berlin-Heidelberg.
Maruri-Aguilar, H., Notari, R., Riccomagno, E., 2007. On the description and identifiability analysis of mixture designs. Statistica Sinica, (1417–1440).
McConkey, B., Mezey, P., Dixon, D., Grenberg, B., 2000. Fractional simplex designs for interaction screening in complex mixtures. Biometrics, 56, 824–832.
Mora, T., Robbiano, L., 1988. The Gröbner fan of an ideal. Journal of Symbolic Computation, 6, 183–208.
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.
Pistone, G., Riccomagno, E., Wynn, H. P., 2001. Algebraic Statistics: Computational Commutative Algebra in Statistics. Chapman&Hall, Boca Raton.
Pistone, G., Rogantin, M., 2007 a. Algebraic statistics of level codings for fractional factorial designs. J. Statist. Plann. Inference 234–244.
Pistone, G., Rogantin, M., 2007 b. Indicator function and complex coding for mixed fractional factorial designs. J. Statist. Plann. Inference 138 (3), 787–802.
Pistone, G., Wynn, H. P., 1996. Generalised confounding with Gröbner bases. Biometrika 83 (3), 653–666.
Scheffé, H., 1958. Experiments with mixtures. J. Roy. Statist. Soc. Ser., B 20, 344–360.
Scheffé, H., 1963. The simplex-centroid design for experiments with mixtures. J. Roy. Statist. Soc. Ser., B 25, 235–263.
Tang, B., Deng, L. Y., 1999. Minimum G 2 -aberration for nonregular fractinal factorial designs. The Annals of Statistics, 27 (6), 1914–1926.
Ye, K. Q., 2003. Indicator function and its application in two-level factorial designs. The Annals of Statistics, 31 (3), 984–994.
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Notari, R., Riccomagno, E. & Rogantin, MP. Two Polynomial Representations of Experimental Design. J Stat Theory Pract 1, 329–346 (2007). https://doi.org/10.1080/15598608.2007.10411844
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DOI: https://doi.org/10.1080/15598608.2007.10411844