Abstract

Many problems of confounding and identifiability for polynomial models in an experimental design can be solved using methods of algebraic geometry. The theory of $Gr\ddot{o}bner$ basis is used to characterize the design. In addition, a fractional factorial design can be uniquely represented by a polynomial indicator function. $Gr\ddot{o}bner$ bases and indicator functions are powerful computational tools to deal with ideals of fractions based on each different theoretical aspects. The problem posed here is to give how to move from one representation to the other. For a given fractional factorial design, the indicator function can be computed from the generating equations in the $Gr\ddot{o}bner$ basis. The theory is tested using some fractional factorial designs aided by a modern computational algebra package CoCoA.

Keywords

• Algebraic geometry;
• Aliased;
• $Gr\ddot{o}bner$ basis;
• indicator function;
• polynomial

• 교락;
• 그뢰브너 기저;
• 다항식;
• 대수기하학;
• 지시함수;

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References

1. Cox, D., Little, J. and O’Shea, D. (1992). Ideal, varieties, and algorithms, Spring-Verlag, New York.
2. Fontana, R., Pistone, G. and Rogantin, M. P. (2000). Classification of two-level factorial fractions. Journal of Statistical Planning and Inference, 87, 149-172. https://doi.org/10.1016/S0378-3758(99)00173-1
3. Park, D. K. and Kim, H. (2003). A New approach for selecting fractional factorial designs. Journal of the Korean Data & Information Science Society, 14, 707-714.
4. Pistone, G., Riccomagno, E. and Rogantin, M. P. (2006). Algebraic statistics method for DOE, Unpublished Manuscript.
5. Pistone, G. and Rogantin, M. P. (2008). Indicator function and complex coding for mixed fractional factorial designs. Journal of Statistical Planning and Inference, 138, 107-121.
6. Pistone, G. and Wynn, H. P. (1996). Generalized confounding with Gr¨obner bases. Biometrika, 83, 653-666. https://doi.org/10.1093/biomet/83.3.653
7. Plackett, R. L. and Burman, J. P. (1946). The design of optimum multifactorial experiments. Biometrika, 33, 305-325. https://doi.org/10.1093/biomet/33.4.305
8. Ye, K. Q. (2003). Indicator functions and its application in two-level factorial designs. Annals of Statistics, 31, 984-994 https://doi.org/10.1214/aos/1056562470
9. Zhang, R. and Park, D. K (2000). Optimal blocking of two-level fractional factorial designs. Journal of Statistical Planning and Inference, 91, 107-121. https://doi.org/10.1016/S0378-3758(00)00133-6