Abstract
Toric models have been recently introduced in the analysis of statistical models for categorical data. The main improvement with respect to classical log-linear models is shown to be a simple representation of structural zeros. In this paper we analyze the geometry of toric models, showing that a toric model is the disjoint union of a number of log-linear models. Moreover, we discuss the connections between the parametric and algebraic representations. The notion of Hilbert basis of a lattice is proved to allow a special representation among all possible parametrizations.
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References
Agresti A. (2002). Categorical Data Analysis, 2nd edn. New York, Wiley
Aoki S., Takemura A. (2005). The largest group of invariance for Markov bases and toric ideals, Technical Report METR 2005-14, Department of Mathematical Informatics. The University of Tokio, Tokio
Bigatti A., Robbiano L. (2001). Toric ideals. Matemática Contemporânea 21, 1–25
Bishop Y.M., Fienberg S., Holland P.W. (1975). Discrete multivariate analysis: theory and practice. Cambridge, MIT Press
CoCoATeam (2004). CoCoA, a system for doing Computations in Commutative Algebra, 4.0 edn, Available at http://cocoa.dima.unige.it.
Diaconis P., Sturmfels B. (1998). Algebraic algorithms for sampling from conditional distributions. Annals of Statistics 26(1): 363–397
Fienberg S. (1980). The Analysis of Cross-Classified Categorical Data. Cambridge, MIT Press
Garcia L.D., Stillman M., Sturmfels B. (2005). ‘Algebraic geometry of Bayesyan networks. Journal of Symbolic Computation 39, 331–355
Geiger D., Heckerman D., King H., Meek C. (2001). Stratified exponential families: graphical models and model selection. Annals of tatistics. 29(3): 505–529
Geiger, D., Meek, C., Sturmfels, B. (2002). On the toric algebra of graphical models. Microsoft Research Report MSR-TR-2002-47.
Goodman L.A. (1979). Multiplicative models for square contingency tables with ordered categories. Biometrika 66(3): 413–418
Hemmecke, R., Hemmecke, R., Malkin, P. (2005). ‘4ti2 version 1.2—computation of Hilbert bases, Graver bases, toric Gröbner bases, and more’, Available at www.4ti2.de.
Kreuzer M., Robbiano L. (2000). Computational Commutative Algebra 1. Berlin Heidelberg New York, Springer
Kreuzer M., Robbiano L. (2005). Computational Commutative Algebra 2. Berlin Heidelberg New York, Springer
Lauritzen S.L. (1996). Graphical Models. New York, Oxford University Press
Pistone G., Wynn H.P. (2003). Statistical toric models. Lecture Notes, Grostat VI, Menton, France
Pistone G., Riccomagno E., Wynn H.P. (2001a). Algebraic Statistics: computational commutative algebra in statistics. Boca Raton, Chapman&Hall/CRC
Pistone, G., Riccomagno, E., Wynn, H. P. (2001b). Computational commutative algebra in discrete statistics. In M. A. G. Viana & D. S. P. Richards (Eds.) Algebraic methods in statistics and probability. Contemporary Mathematics, (pp. 267–282) American Mathematical Society, Vol. 287.
Rapallo F. (2003). Algebraic Markov bases and MCMC for two-way contingency tables. Scandinavian Journal of Statistics 30(2): 385–397
Sturmfels B. (1993). Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation. Berlin Heidelberg New York, Springer
Sturmfels, B. (1996). Gröbner bases and convex polytopes. In University lecture series (Providence, R.I.), Vol. 8 American Mathematical Society
Sturmfels, B. (2002). Solving systems of polynomial equations, In CBMS Regional Conference Series in Mathematics, Vol. 97 American Mathematical Society, Providence, RI.
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Rapallo, F. Toric statistical models: parametric and binomial representations. AISM 59, 727–740 (2007). https://doi.org/10.1007/s10463-006-0079-z
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DOI: https://doi.org/10.1007/s10463-006-0079-z