Tropical Geometric Variation of Tree Shapes

Abstract

We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree’s branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.

Tropical medians by transportation

“…There is a considerable amount of work on the tropical metric geometry of the space of equidistant trees [27-29, 36, 43]. This is quite natural, because the symmetric tropical distance between two equidistant trees can be interpreted naturally, as the cost of changing one tree into the other along a tropical line segment, where the cost is measured in the ∞ norm; see [27] for details. Here we study the asymmetric tropical distance, which has a similar interpretation, but for the 1 norm.…”

Estimating the mean in the space of ranked phylogenetic trees

“…To overcome these issues in a statistically sound way, Holmes et al [16] initiated a research programme aimed at developing a mathematical framework for statistical analyses over the space of phylogenetic trees. The programme has since received significant attention in the literature with a number of promising treespaces introduced [17][18][19][20][21][22][23][24][25][26][27] and a range of statistical methods developed [28][29][30][31][32][33][34][35][36][37][38]. However, the ultimate milestone of the original programme of introducing fundamental statistics, such as the mean, variance, confidence intervals, for probability distributions over any treespace in a way that would enable a wide range of practical applications remains illusive [23,[39][40][41].…”

“…These results exposed the main complications that need to be overcome to enable statistically sound and practically useful tree inference methods. These include the stickiness of the mean [41] (irresponsiveness of the mean tree to changes in the tree sample), prevalence of star-like mean trees even for phylogenetically informative data [21,41], stickiness of the first principal component [25,29], the high dimensionality problem for small samples of trees [23], and others [34]. Some of these complications are actively being addressed with, e.g.…”