Markov chains, quotient ideals and connectivity with positive margins

Algebraic and Geometric Methods in Statistics

Volumn , Issue , 2009, Pages 99-110

  Chen, Yuguo ^a   Dinwoodie, Ian H ^b   Yoshida, Ruriko ^c  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b DUKE UNIVERSITY   (United States)

^c UNIVERSITY OF KENTUCKY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77958068041     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1017/CBO9780511642401.006     Document Type: Chapter

References (31)

1. 4ti2 Team (2006). 4ti2 - A software package for algebraic, geometric and combinatorial problems on linear spaces (available at www.4ti2.de).
2. Bayer, D., Popescu, S., and Sturmfels, B. (2001). Syzygies of unimodular Lawrence ideals, Journal fur die reine und angewandte Mathematik 534, 169-86.
3. Besag, J., and Clifford, P. (1989). Generalized Monte Carlo significance tests, Biometrika 76, 633-42.
4. Booth, J. G., and Butler, J. W. (1999). An importance sampling algorithm for exact conditional tests in loglinear models, Biometrika 86, 321-32.
5. Bunea, F., and Besag, J. (2000). MCMC in I x J x K contingency tables, Fields Institute Communications 26, 23-36.
6. Caffo, B. (2006). exactLoglinTest: A Program for Monte Carlo Conditional Analysis of Log-linear Models (available at www.cran.r-project.org).
7. Chen, Y., Dinwoodie, I. H., Dobra, A. and Huber, M. (2005). Lattice points, contingency tables and sampling. In Contemporary Mathematics. Barvinok, A., Beck, M., Haase, C., Reznick, B., and Welker, V. eds. (American Mathematical Society Vol. 374) 65-78.
8. Chen, Y., Dinwoodie, I. H., and MacGibbon, B. (2007). Sequential importance sampling for case-control data, Biometrics 63(3), 845-55.
9. Chen, Y., Dinwoodie, I. H., and Sullivant, S. (2006). Sequential importance sampling for multiway tables, Annals of Statistics 34, 523-45.
10. Christensen, R. (1990). Log-Linear Models (New York, Springer-Verlag).
11. CoCoATeam (2007). CoCoA, a system for doing Computations in Commutative Algebra, 4.7 edn (available at http://cocoa.dima.unige.it).
12. De Loera, J. and Onn, S. (2005). Markov bases of three-way tables are arbitrarily complicated, Journal of Symbolic Computation 41, 173-81.
13. Diaconis, P. and Sturmfels, B. (1998). Algebraic methods for sampling from conditional distributions, Annals of Statistics 26, 363-97.
14. Diaconis, P., Eisenbud, D., and Sturmfels, B. (1998). Lattice walks and primary decomposition. In Mathematical Essays in Honor of Gian-Carlo Rota, Sagan, B. E. and Stanley, R. P. eds. (Boston, Birkhauser) 173-93.
15. Fallin, D., Cohen, A., Essioux, L., Chumakov, I., Blumenfeld, M., Cohen, D., and Schork, N. J. (2001). Genetic analysis of case/control data using estimated haplotype frequencies: application to APOE locus variation and Alzheimer’s disease, Genome Research 11, 143-51.
16. Geiger, D., Meek, C., and Sturmfels, B. (2006). On the toric algebra of graphical models, Annals of Statistics 34, 1463-92.
17. Grayson, D. and Stillman, M. (2006). Macaulay 2, a software system for research in algebraic geometry (available at www.math.uiuc.edu/Macaulay2/).
18. Greuel, G.-M., Pfister, G. and Schonemann, H. (2005). SINGULAR 3.0. A Computer Algebra, System for Polynomial Computations. Centre for Computer Algebra (available at www.singular.uni-kl.de).
19. Guo, S. W., and Thompson, E. A. (1992). Performing the exact test of Hardy-Weinberg proportion for multiple alleles, Biometrics 48, 361-72.
20. Hemmecke, R., and Malkin, P. (2005). Computing generating sets of lattice ideals (available at arXiv:math.CO/0508359).
21. Hosten, S., and Shapiro, J. (2000). Primary decomposition of lattice basis ideals, Journal of Symbolic Computation 29, 625-39.
22. Hosten, S., and Sullivant, S. (2002). Grobner basis and polyhedral geometry of reducible and cyclic models, Journal of Combinatorial Theory A 100, 277-301.
23. Hosten, S., and Sullivant, S. (2004). Ideals of adjacent minors, Journal of Algebra 277, 61542.
24. Kreuzer, M., and Robbiano, L. (2000). Computational Commutative Algebra (New York, Springer-Verlag).
25. Pistone, G., and Wynn, H. (1999). Finitely generated cumulants, Statistica Sinica 9(4), 1029-52.
26. Pistone, G., Riccomagno, E. and Wynn, H. P. (2001). Algebraic Statistics (Boca Raton, Chapman & Hall/cRc).
27. R Development Core Team (2004). R: A Language and Environment for Statistical Computing (available at www.R-project.org).
28. Schrijver, A. (1989). Theory of linear and integer programming (Chichester, John Wiley & Sons).
29. Sturmfels, B. (1996). Grobner Bases and Convex Polytopes (Providence, RI, American Mathematical Society).
30. Sturmfels, B. (2002). Solving Systems of Polynomial Equations (Providence, RI, American Mathematical Society).
31. Whittaker, J. (1990). Graphical Models in Applied Mathematical Multivariate Statistics (Chichester, John Wiley & Sons).