ALGEBRAIC STATISTICS of DESIGN EXPERIMENTS Algebra Has Many
ALGEBRAIC STATISTICS OF DESIGN EXPERIMENTS
GIOVANNI PISTONE
Algebra has many application in Statistics. The keyword Algebraic Statistics denotes the application to Statistics of polynomial algebra. In particular, it was observed in [6] that the set of treatments in a designed experiments is the set of solutions of a system of polynomial equations and tools from modern Commutative Algebra [1] are useful in discussing problems of interest in Design of Experiments (DoE). This idea was developed in [5]. Applications to statistical models were previously discussed in [2]. A tutorial presentation of algebraic DoE is [4], while Part II Designed Experiments of [3] contains an overview of recent research. We present the algebraic basics (ideal, variety, basis, Gr¨obnerbasis) and some examples of application to DoE.
References [1] David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992, An introduction to computational algebraic geometry and commutative algebra. MR 1189133 (93j:13031) [2] Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist. 26 (1998), no. 1, 363–397. MR 99j:62137 [3] Paolo Gibilisco, Eva Riccomagno, Maria Piera Rogantin, and Henry P. Wynn (eds.), Algebraic and geometric methods in statistics, Cambridge University Press, Cambridge, 2010. MR 2640515 (2011a:62007) [4] Giovanni Pistone, Eva Riccomagno, and Maria Piera Rogantin, Methods in algebraic statistics for the design of experiments, Optimal Design and Related Areas in Optimization and Statistics (Luc Pronzato and Antony A. Zigljavsky, eds.), Springer Optimization and its Applications, no. 28, Springer-Verlag, 2009, pp. 97–132. [5] Giovanni Pistone, Eva Riccomagno, and Henry P. Wynn, Algebraic statis- tics, Monographs on Statistics and Applied Probability, vol. 89, Chapman & Hall/CRC, Boca Raton, FL, 2001, Computational commutative algebra in sta- tistics. MR 2332740 (2008f:62098) [6] Giovanni Pistone and Henry P. Wynn, Generalised confounding with Gr¨obner bases, Biometrika 83 (1996), no. 3, 653–666. MR 1 423 881
Collegio Carlo Alberto, Moncalieri, Italy E-mail address: [email protected]
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