Limiting behaviour of Fréchet means in the space of phylogenetic trees

Barden, D.; Le, Huiling; Owen, M.

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.

Citation

Barden, D., Le, H., & Owen, M. (2018). Limiting behaviour of Fréchet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics, 70(1), 99-129. https://doi.org/10.1007/s10463-016-0582-9

• Journal Article Type: Article
• Acceptance Date: Sep 12, 2016
• Online Publication Date: Oct 19, 2016
• Publication Date: Feb 1, 2018
• Deposit Date: Nov 22, 2016
• Publicly Available Date: Nov 22, 2016
• Journal: Annals of the Institute of Statistical Mathematics
• Print ISSN: 0020-3157
• Electronic ISSN: 1572-9052
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 70
• Issue: 1
• Pages: 99-129
• DOI: https://doi.org/10.1007/s10463-016-0582-9
• Keywords: Central limit theorem, Fréchet mean, Log map, Phylogenetic trees, Stratified manifold
• Public URL: https://nottingham-repository.worktribe.com/output/908832
• Publisher URL: http://link.springer.com/article/10.1007%2Fs10463-016-0582-9
• Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s10463-016-0582-9
• Contract Date: Nov 22, 2016