Article
Full entry |
PDF
(0.9 MB)
Feedback
PDF
(0.9 MB)
Feedback
Keywords:
Brownian motion tree model; ultrametric matrices; toric geometry
Brownian motion tree model; ultrametric matrices; toric geometry
Summary:
Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.
References:
[1] Anderson, T. W.: Estimation of covariance matrices which are linear combinations or whose inverses are linear combinations of given matrices. In: Essays in Probability and Statistics (I.|,M. Mahalanobis, P. C. Rao, C. R. Bose, R. C. Chakravarti and K. J. C. Smith, eds.), Univ. of North Carolina Press, Chapel Hill, 1970, pp. 1-24. MR 0277057
[2] Bossinger, L., Fang, X., Fourier, G., Hering, M., Lanini, M.: Toric degenerations of Gr(2,n) and Gr(3,6) via plabic graphs. Ann. Combinator. 22 (2018), 3, 491-512. DOI 10.1007/s00026-018-0395-z | MR 3845745
[3] Buneman, P.: The recovery of trees from measures of dissimilarity. In: Mathematics in the Archaeological and Historical Sciences (F. Hodson et al., ed.), Edinburgh University Press, 1971, pp. 387-395.
[4] Carlson, D., Markham, T. L.: Schur complements of diagonally dominant matrices. Czechosl. Math. J. 29 (1979), 2, 246-251. DOI 10.21136/CMJ.1979.101601 | MR 0529512
[5] Dellacherie, C., Martinez, S., Martin, J. San: Inverse M-matrices and ultrametric matrices. Springer 2118, 2014. DOI 10.1007/978-3-319-10298-6_1 | MR 3289211
[6] Draisma, J., Kuttler, J.: On the ideals of equivariant tree models. Math. Ann. 344 (2009), 3, 619-644. DOI 10.1007/s00208-008-0320-6 | MR 2501304
[7] Sullivant, J. S., Talaska, K.: Positivity for Gaussian graphical models. Adv. Appl. Math. 50 (2013), 5, 661-674. DOI 10.1016/j.aam.2013.03.001 | MR 3044565
[8] Felsenstein, J.: Maximum-likelihood estimation of evolutionary trees from continuous characters. Amer. J. Human Genetics 25 (1973), 5, 471-492.
[9] Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry.
[10] Kaveh, K., Manon, Ch.: Khovanskii bases, higher rank valuations and tropical geometry. SIAM J. Appl. Algebra Geometry 3 (2019), 2, 292-336. DOI 10.1016/j.aam.2013.03.001 | MR 3949692
[11] Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. American Mathematical Society, Graduate Studies in Mathematics 161, Providence 2015. DOI 10.1090/gsm/161 | MR 3287221
[12] Michałek, M., Sturmfels, B., Uhler, C., Zwiernik, P.: Exponential varieties. Proc. London Math. Soc. (3), 112 (2016), 1, 27-56. DOI 10.1112/plms/pdv066 | MR 3458144
[13] Semple, Ch., Steel, M.: Phylogenetics. Oxford University Press, 2003. DOI 10.1080/10635150490888895 | MR 2060009
[14] Moulton, V., Steel, M.: Peeling phylogenetic ‘oranges’. Adv. App. Mathemat. 33 (2004), 4, 710-727. DOI 10.1016/j.aam.2004.03.003 | MR 2095862
[15] Sullivant, S., Talaska, K., Draisma, J.: Trek separation for Gaussian graphical models. Ann. Stat. 38 (2010), 3, 1665-1685. DOI 10.1214/09-aos760 | MR 2662356
[16] Varga, R. S., Nabben, R.: On symmetric ultrametric matrices. Numerical Linear Algebra (L. Reichel et al., eds.), de Gruyter, New York 1993, pp. 193-199. DOI 10.1515/9783110857658.193 | MR 1244160
[17] Zwiernik, P., Uhler, C., Richards, D.: Maximum likelihood estimation for linear Gaussian covariance models. J. Roy. Stat. Soc.: Series B (Stat. Method.) 79 (2017), 4, 1269-1292. DOI 10.1111/rssb.12217 | MR 3689318
Search
Partner of
