Central limit theorems for Fréchet means in the space of phylogenetic trees

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

“…This form may change as x varies within an orthant. The following example illustrates this feature in the space X 2 in R 5 , which was called Q 5 in [2], consisting of the five orthants depicted in Figure 2, where all five axes are mutually orthogonal. Although x lies in the same orthant in the second and third cases, the forms for Φ(x; x * ), as a function of x, differ in the two cases.…”

“…The intrinsic metric on X m is the length metric as defined in [6]. It is the metric d for which, for any two points x 1 and x 2 in X m , the distance d(x 1 , x 2 ) is the infimum of the lengths of piecewise linear paths in X m joining x 1 to x 2 . In particular, a geodesic will also be piecewise linear and linear within each stratum.…”

“…. , O k ) of the geodesic from x 1 to x 2 will, naturally, depend on both x 1 and x 2 . If x 1 lies in a top-dimensional stratum it will have m strictly positive coordinates, all of which, assuming that none are also positive in x 2 , must become zero somewhere along the geodesic and at least one must become zero on each change of stratum as they cannot vanish within a stratum of the carrier.…”

“…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 .…”