arXiv subjects
Publications and source records attributed to Dennis Barden.
The first part of the paper studies the expression for, and the properties of, the logarithm map on an orthant space, which is a simple stratified space, with the aim of analysing Frechet means of probability measures on such a space. In the second part, we use these results to characterise Frechet means and to derive various of their properties, including the limiting distribution of sample Frechet means.
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 Fréchet 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 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.