Limiting Behaviour of Fr\'echet Means in the Space of Phylogenetic Trees

TL;DR

This paper extends the analysis of Fréchet means from the space of four-leaf phylogenetic trees to spaces with five or more leaves, characterizing their limiting behavior and distributions based on the topological strata.

Contribution

It introduces a generalized log map for ${oldsymbol T}_{m}$ and characterizes Fréchet means and their limiting distributions in higher-dimensional phylogenetic tree spaces.

Findings

Limiting distributions depend on the strata's co-dimensions.

Generalization of the log map to ${oldsymbol T}_{m}$.

Characterization of Fréchet means in top-dimensional and co-dimension one strata.

Abstract

As demonstrated in our previous work on ${\boldsymbol T}_{4}$, the space of phylogenetic trees with four leaves, the global, as well as the local, topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fr\'echet means of a probability distribution on ${\boldsymbol T}_{4}$. Nevertheless, the techniques used in that paper were specific to ${\boldsymbol T}_{4}$ and cannot be adapted to analyse Fr\'echet means in the space ${\boldsymbol T}_{m}$ of phylogenetic trees with $m(\geqslant5)$ leaves. To investigate the latter, this paper first studies the log map of ${\boldsymbol T}_{m}$, a generalisation of the inverse of the exponential map on a Riemannian manifold. Then, in terms of a modified version of the log map, we characterise Fr\'echet means in ${\boldsymbol T}_{m}$ that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Fr\'echet 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 Fr\'echet means lie.