Limiting behaviour of Fréchet means in the space of phylogenetic trees

Abstract

As demonstrated in our previous work on 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 T4. Nevertheless, the techniques used in that paper were specific to T4 and cannot be adapted to analyse Fréchet means in the space Tm of phylogenetic trees with m( 5) leaves. To investigate the latter, this paper first studies the log map of Tm, 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 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.

Persistent homology analysis of brain artery trees

“…For instance, Feragen et al (2011) developed an approach that avoids both the planar embedding and fixed-leaf-set problems, and Nye (2011) invented an analogue of principal component analysis for phylogenetic trees. Hotz et al (2013), followed by Barden, Le and Owen (2013, 2014), investigated surprising nonstandard central limit theory in phylogenetic tree spaces. Finally, an analysis [Wright et al (2013)] of a different, and smaller, set of MRA brain artery images also found a connection between vessel length and healthy aging.…”

The logarithm map, its limits and Fréchet means in orthant spaces

“…Although we do not make it explicit, in view of our previous results for T m+2 and the comments in [15], our interest in this paper is primarily in the case that x * lies in a stratum of local co-dimension at least two. The results, when restricted to a locally top-dimensional or co-dimension one stratum, do generalise those for tree spaces in [3] although the approach here is necessarily more complex in order to encompass all cases.…”

“…With the description of the carrier, as well as the results on the support, of a geodesic in the previous section, we are now in a position to derive and analyse its initial tangent vector, or equivalently log x * (x) for x * ∈ σ = O(E). As in [2,3] for the space of trees, our analysis will mainly involve a modified version of the logarithm map. For this, since the tangent cones at various points in σ are all parallel, we may parallel translate them to the cone point o, the origin in R M , to produce a common isometric copy C σ .…”

“…Note that, although the origin corresponds to the cone point o of the orthant space X m , C σ is not the tangent cone to X m at o, neither being contained in the other, unless σ = {o}. Note also that, when X m is a tree space and x * lies in a top-dimensional stratum, Φ(x; x * ) was called the modified logarithm map and was denoted by Φ x * (x) in [3], and the permutation map π there corresponds to the linear transformation j given by Definition 5.…”

“…On the other hand, an algorithm for finding the geodesic between any two given trees in the tree space T m+2 was given in [19] and, using the analysis behind that algorithm, the expression for the logarithm map log x * was obtained in [3] when x * lies in a top-dimensional stratum. Although this expression for log x * could be extended to more general orthant spaces, it is noted in [3] that these results are not adequate to provide a tool for analysing Fréchet means when they lie in any stratum of co-dimension at least two. The latter requires a better understanding of the behaviour of the logarithm map as the end points of the geodesics move within and between strata.…”