Identifiability of two-tree mixtures for group-based models

IEEE/ACM Transactions on Computational Biology and Bioinformatics

Volumn 8, Issue 3, 2011, Pages 710-722

  Allman, Elizabeth S ^a   Petrović, Sonia ^b   Rhodes, John A ^a   Sullivant, Seth ^c  

^a UNIVERSITY OF ALASKA FAIRBANKS   (United States)

^b UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^c NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

group based model; identifiability of phylogenetic models; Phylogenetic mixture

Indexed keywords

4-STATE; ALGEBRAIC TECHNIQUES; BIOLOGICAL MECHANISMS; DNA SEQUENCE DATA; GROUP-BASED; GROUP-BASED MODEL; HORIZONTAL GENE TRANSFER; IDENTIFIABILITY; MIXTURE MODEL; PHYLOGENETIC MIXTURE; SYMMETRIC MODEL; TREE TOPOLOGY;

GENE TRANSFER; GENES; MIXTURES; TREES (MATHEMATICS);

TOPOLOGY;

DNA;

ALGORITHM; ARTICLE; BIOLOGICAL MODEL; BIOLOGY; GENETICS; METHODOLOGY; NUCLEIC ACID DATABASE; PHYLOGENY; PROBABILITY; STATISTICAL MODEL;

ALGORITHMS; COMPUTATIONAL BIOLOGY; DATABASES, NUCLEIC ACID; DNA; MARKOV CHAINS; MODELS, GENETIC; MODELS, STATISTICAL; PHYLOGENY;

EID: 79952860539     PISSN: 15455963     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCBB.2010.79     Document Type: Article

References (35)

1. E. S. Allman, C. Ané, and J. A. Rhodes, "Identifiability of a Markovian Model of Molecular Evolution with Gamma-Distributed Rates, " Advances in Applied Probability, vol. 40, pp. 229-249, arXiv:0709.0531, 2008. (Pubitemid 351772466)
2. E. S. Allman, C. Matias, and J. A. Rhodes, "Identifiability of Parameters in Latent Structure Models with Many Observed Variables, " Annals of Statistics, vol. 37, no. 6A, pp. 3099-3132, 2009.
3. E. S. Allman, S. Petrovic, J. A. Rhodes, and S. Sullivant, "Supplementary Material, " http://www4.ncsu.edu~smsulli2/Pubs/ TwoTreesWebsite/twotrees.html, 2009.
4. E. S. Allman and J. A. Rhodes, "Phylogenetic Invariants for the General Markov Model of Sequence Mutation, " Math. Bioscience, vol. 186, no. 2, pp. 113-144, 2003. (Pubitemid 37414196)
5. E. S. Allman and J. A. Rhodes, "The Identifiability of Tree Topology for Phylogenetic Models, Including Covarion and Mixture Models, " J. Computational Biology, vol. 13, no. 5, pp. 1101-1113, 2006. (Pubitemid 43993453)
6. E. S. Allman and J. A. Rhodes, "Phylogenetic Ideals and Varieties for the General Markov Model, " Advances in Applied Math., vol. 40, no. 2, pp. 127-148, 2008. (Pubitemid 351199916)
7. E. S. Allman and J. A. Rhodes, "The Identifiability of Covarion Models in Phylogenetics, " IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 6, no. 1, pp. 76-88, Jan.-Mar. 2009.
8. J. A. Cavender and J. Felsenstein, "Invariants of Phylogenies in a Simple Case with Discrete States, " J. Classification, vol. 4, pp. 57-71, 1987.
9. J. T. Chang, "Full Reconstruction of Markov Models on Evolutionary Trees: Identifiability and Consistency, " Math. Bioscience, vol. 137, no. 1, pp. 51-73, 1996. (Pubitemid 26307662)
10. D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, second ed. Springer-Verlag, 1997.
11. J. Draisma, "A Tropical Approach to Secant Dimensions, " J. Pure and Applied Algebra, vol. 212, no. 2, pp. 349-363, 2008.
12. M. Drton, B. Sturmfels, and S. Sullivant, "Lectures on Algebraic Statistics, " Oberwolfach Seminars, vol. 39, Birkhauser Basel, 2008.
13. S. N. Evans and T. P. Speed, "Invariants of Some Probability Models Used in Phylogenetic Inference, " Annals of Statistics, vol. 21, no. 1, pp. 355-377, 1993.
14. W. Fulton, "The William H. Roever Lectures in Geometry, " Introduction to Toric Varieties, Princeton Univ. Press, 1993.
15. D. R Grayson and M. E. Stillman, "Macaulay2, a Software System for Research in Algebraic Geometry, " http://www.math.uiuc. edu/Macaulay2/, 2002.
16. G.-M. Greuel, G Pfister, and H. Schonemann, "SINGULAR 3.1.0 A Computer Algebra System for Polynomial Computations, " http://www.singular. uni-kl.de, 2009.
17. J. Harris, Algebraic Geometry: A First Course. Springer-Verlag, 1992.
18. M. D. Hendy, "The Relationship between Simple Evolutionary Tree Models and Observable Sequence Data, " Systematic Zoology, vol. 38, pp. 310-321, 1989.
19. M. D. Hendy and D. Penny, "Spectral Analysis of Phylogenetic Data, " J. Classification, vol. 10, pp. 1-20, 1993.
20. M. D. Hendy and D. Penny, "Complete Families of Linear Invariants for Some Stochastic Models of Sequence Evolution, with and without the Molecular Clock Assumption, " J. Computational Biology, vol. 3, no. 1, pp. 19-31, 1996. (Pubitemid 26182761)
21. S. Hoşten, A. Khetan, and B. Sturmfels, "Solving The Likelihood Equations, " Foundations of Computational Math., vol. 5, pp. 389-407, arXiv:math. ST/0408270, 2005. (Pubitemid 43183915)
22. J. A. Lake, "A Rate Independent Technique for Analysis of Nucleic Acid Sequences: Evolutionary Parsimony, " Molecular Biology and Evolution, vol. 4, no. 2, pp. 167-191, 1987. (Pubitemid 17054810)
23. F. A. Matsen, E. Mossel, and M. Steel, "Mixed-Up Trees: The Structure of Phylogenetic Mixtures, " Bull. of Math. Biology, vol. 70, no. 4, pp. 1115-1139, 2008.
24. F. A. Matsen and M. A. Steel, "Phylogenetic Mixtures on a Single Tree Can Mimic a Tree of Another Topology, " Systematic Biology, vol. 56, no. 5, pp. 767-775, 2007.
25. E. Mossel and E. Vigoda, "Phylogenetic MCMC Algorithms Are Misleading on Mixtures of Trees, " Science, vol. 309, pp. 2207-2209, 2005. (Pubitemid 41396068)
26. S. Rudich, "Complexity Theory: From Godel to Feynman, " Computational Complexity Theory, vol. 10, pp. 5-87, Am. Math. Soc, 2004.
27. C. Semple and M. Steel, Phylogenetics, vol. 24. Oxford Univ. Press, 2003.
28. D. Speyer and B. Sturmfels, "The Tropical Grassmannian, " Advances in Geometry, vol. 4, no. 3, pp. 389-411, 2004.
29. M. A. Steel and Y. X. Fu, "Classifying and Counting Linear Phylogenetic Invariants for the Jukes-Cantor Model, " J. Computational Biology, vol. 2, no. 1, pp. 39-47, 1995.
30. B. Sturmfels, Grobner Bases and Convex Polytopes, vol. 8. Am. Math. Soc, 1996.
31. B. Sturmfels and S. Sullivant, "Toric Ideals of Phylogenetic Invariants, " J. Computational Biology, vol. 12, no. 2, pp. 204-228, 2005. (Pubitemid 40444588)
32. L. Szekely, P. L. Erdos, M. A. Steel, and D. Penny, "A Fourier Inversion Formula for Evolutionary Trees, " Applied Math. Letters, vol. 6, no. 2, pp. 13-17, 1993.
33. L. A. Szekely, M. A. Steel, and P. L. Erdos, "Fourier Calculus on Evolutionary Trees, " Advances in Applied Math., vol. 14, no. 2, pp. 200-210, 1993.
34. D. Stefankovic and E. Vigoda, "Phylogeny of Mixture Models: Robustness of Maximum Likelihood and Non-Identifiable Distributions, " J. Computational Biology, vol. 14, no. 2, pp. 156-189, 2007. (Pubitemid 46709353)
35. J. Chai and E. A. Housworth, "On Rogers's Proof of Identifiability for the GTR + Gamma + I Model, " to appear in Systematic Biology.