Central limit theorems for Fréchet means in the space of phylogenetic trees

Dennis Barden (University of Cambridge)
Huiling Le (University of Nottingham)
Megan Owen (University of Waterloo)

Abstract


This paper studies the characterisation, and the limiting distributions, of Fréchet means in the space of phylogenetic trees. This space is topologically stratified, as well as being a CAT(0) space. We use a generalised version of the Delta method to demonstrate non-classical behaviour arising from the global topological structure of the space. In particular, we show that, for the space of trees with four leaves, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fréchet means lie.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-25

Publication Date: February 16, 2013

DOI: 10.1214/EJP.v18-2201

References

  • Basrak, Bojan. Limit theorems for the inductive mean on metric trees. J. Appl. Probab. 47 (2010), no. 4, 1136--1149. MR2752884
  • Bhattacharya, Rabi; Patrangenaru, Vic. Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 (2003), no. 1, 1--29. MR1962498
  • Bhattacharya, Rabi; Patrangenaru, Vic. Large sample theory of intrinsic and extrinsic sample means on manifolds. II. Ann. Statist. 33 (2005), no. 3, 1225--1259. MR2195634
  • Billera, Louis J.; Holmes, Susan P.; Vogtmann, Karen. Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 (2001), no. 4, 733--767. MR1867931
  • Bridson, Martin R.; Haefliger, André. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. xxii+643 pp. ISBN: 3-540-64324-9 MR1744486
  • Dryden, I. L.; Mardia, K. V. Statistical shape analysis. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1998. xx+347 pp. ISBN: 0-471-95816-6 MR1646114
  • Felsenstein, J.: Confidence limits on phylogenies: an approach using the bootstrap. Evolution 39 (1985), 783--791.
  • Hendriks, Harrie; Landsman, Zinoviy. Asymptotic behavior of sample mean location for manifolds. Statist. Probab. Lett. 26 (1996), no. 2, 169--178. MR1381468
  • Hendriks, Harrie; Landsman, Zinoviy. Mean location and sample mean location on manifolds: asymptotics, tests, confidence regions. J. Multivariate Anal. 67 (1998), no. 2, 227--243. MR1659156
  • Holmes, Susan. Bootstrapping phylogenetic trees: theory and methods. Silver anniversary of the bootstrap. Statist. Sci. 18 (2003), no. 2, 241--255. MR2026083
  • Holmes, S.: Statistics for phylogenetic trees. Theoretical Population Biology 63 (2003), 17--32.
  • Holmes, S.: Statistical approach to tests involving phylogenies. In: Mathematics of Evolution and Phylogeny, O. Gascuel Editor. Oxford University Press, USA, 2007.
  • Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., Nolen, J., Owen, M., Patrangenaru, V. and Skwerer, S.: Sticky central limit theorems on open books. Ann. Appl. Probab. to appear, (2012). arXiv:1202.4267.
  • Goresky, Mark; MacPherson, Robert. Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 14. Springer-Verlag, Berlin, 1988. xiv+272 pp. ISBN: 3-540-17300-5 MR0932724
  • Kendall, D. G.; Barden, D.; Carne, T. K.; Le, H. Shape and shape theory. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1999. xii+306 pp. ISBN: 0-471-96823-4 MR1891212
  • Kendall, Wilfrid S.; Le, Huiling. Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables. Braz. J. Probab. Stat. 25 (2011), no. 3, 323--352. MR2832889
  • Miller, E., Owen, M. and Provan, S.: Averaging metric phylogenetic trees. In preparation, (2012).
  • Owen, Megan. Computing geodesic distances in tree space. SIAM J. Discrete Math. 25 (2011), no. 4, 1506--1529. MR2873201
  • Owen, M. and Provan, J. S.: A fast algorithm for computing geodesic distances in tree space. IEEE/ACM Trans. Computational Biology and Bioinformatics 8 (2011), 2--13.
  • Sturm, Karl-Theodor. Probability measures on metric spaces of nonpositive curvature. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 357--390, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003. MR2039961
  • Vogtmann, K. (2007) Geodesics in the space of trees, (2007), www.math.cornell.edu/simvogtmann/papers/TreeGeodesicss/index.html.
  • Ziezold, Herbert. On expected figures and a strong law of large numbers for random elements in quasi-metric spaces. Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians (Tech. Univ. Prague, Prague, 1974), Vol. A, pp. 591--602. Reidel, Dordrecht, 1977. MR0501230
  • Z. Yang, Z. and Rannala, B.: (1997) Bayesian phylogenetic inference using DNA sequences: a Markov chain Monte Carlo method. Molecular Biology and Evolution 14 (1997), 717-724.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.