Limiting Behaviour of Fréchet Means in the Space of Phylogenetic Trees

2014/09/26 by Barden, Dennis, Le, Huiling, Owen, Megan
#60D05 #60F05 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1409.7602

Abstract

As demonstrated in our previous work on \boldsymbol T₄, 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 Fréchet means of a probability distribution on \boldsymbol T₄. Nevertheless, the techniques used in that paper were specific to \boldsymbol T₄ and cannot be adapted to analyse Fréchet 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 Fréchet means in \boldsymbol T_m 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.

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