Limiting behaviour of Fréchet means in the space of phylogenetic trees

Barden, D. and Le, Huiling and Owen, M. (2016) Limiting behaviour of Fréchet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics . ISSN 1572-9052

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Abstract

As demonstrated in our previous work on T4, the space of phylogenetic trees with four leaves, the topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fréchet means in T4. Nevertheless, the techniques used in that paper cannot be adapted to analyse Fréchet means in the space Tm of phylogenetic trees with m(⩾5)m(⩾5) leaves. To investigate the latter, this paper first studies the log map of Tm. Then, in terms of a modified version of this map, we characterise Fréchet means in Tm that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Fréchet means, generalising our previous results. In particular, the results show that, 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.

Item Type: Article
Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s10463-016-0582-9
Keywords: Central limit theorem, Fréchet mean, Log map, Phylogenetic trees, Stratified manifold
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: https://doi.org/10.1007/s10463-016-0582-9
Depositing User: Eprints, Support
Date Deposited: 22 Nov 2016 09:44
Last Modified: 22 Nov 2016 09:48
URI: http://eprints.nottingham.ac.uk/id/eprint/38874

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