TY - JOUR

T1 - Perturbation method for determining the group of invariance of hierarchical models

AU - Sei, Tomonari

AU - Aoki, Satoshi

AU - Takemura, Akimichi

PY - 2009/10

Y1 - 2009/10

N2 - We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in [S. Aoki, A. Takemura, The largest group of invariance for Markov bases and toric ideals, J. Symbolic Comput. 43 (5) (2008) 342-358]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof by Aoki and Takemura which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.

AB - We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in [S. Aoki, A. Takemura, The largest group of invariance for Markov bases and toric ideals, J. Symbolic Comput. 43 (5) (2008) 342-358]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof by Aoki and Takemura which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.

KW - Computational algebraic statistics

KW - Group action

KW - Stabilizer

KW - Sudoku

KW - Wreath product

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U2 - 10.1016/j.aam.2009.02.005

DO - 10.1016/j.aam.2009.02.005

M3 - Article

AN - SCOPUS:69849087694

VL - 43

SP - 375

EP - 389

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 4

ER -