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Minimal Basis for a Connected Markov Chain over 3 × 3 × K Contingency Tables with Fixed Two‐Dimensional Marginals
Author(s) -
Aoki Satoshi,
Takemura Akimichi
Publication year - 2003
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00278
Subject(s) - markov chain , contingency table , mathematics , basis (linear algebra) , variable order markov model , discrete mathematics , chain (unit) , computation , sampling (signal processing) , construct (python library) , markov model , algorithm , combinatorics , statistics , computer science , physics , geometry , filter (signal processing) , astronomy , computer vision , programming language
This paper considers a connected Markov chain for sampling 3 × 3 × K contingency tables having fixed two‐dimensional marginal totals. Such sampling arises in performing various tests of the hypothesis of no three‐factor interactions. A Markov chain algorithm is a valuable tool for evaluating P ‐values, especially for sparse datasets where large‐sample theory does not work well. To construct a connected Markov chain over high‐dimensional contingency tables with fixed marginals, algebraic algorithms have been proposed. These algorithms involve computations in polynomial rings using Gröbner bases. However, algorithms based on Gröbner bases do not incorporate symmetry among variables and are very time‐consuming when the contingency tables are large. We construct a minimal basis for a connected Markov chain over 3 × 3 × K contingency tables. The minimal basis is unique. Some numerical examples illustrate the practicality of our algorithms.