A 'stochastic safety radius' for distance-based tree reconstruction

TL;DR

This paper introduces a stochastic safety radius concept for evaluating the accuracy of distance-based tree reconstruction methods, providing bounds and simulation-based comparisons that explain the high performance of Neighbor-Joining.

Contribution

It establishes bounds on the stochastic safety radius for tree reconstruction, offering a new framework to compare and understand method performance.

Findings

Neighbor-Joining has a stochastic safety radius close to optimal.

The stochastic safety radius provides a new performance comparison metric.

Bounds on the stochastic safety radius are mathematically derived.

Abstract

A variety of algorithms have been proposed for reconstructing trees that show the evolutionary relationships between species by comparing differences in genetic data across present-day taxa. If the leaf-to-leaf distances in a tree can be accurately estimated, then it is possible to reconstruct this tree from these estimated distances, using polynomial-time methods such as the popular `Neighbor-Joining' algorithm. There is a precise combinatorial condition under which distance-based methods are guaranteed to return a correct tree (in full or in part) based on the requirement that the input distances all lie within some `safety radius' of the true distances. Here, we explore a stochastic analogue of this condition, and mathematically establish upper and lower bounds on this `stochastic safety radius' for distance-based tree reconstruction methods. Using simulations, we show how this notion provides a new way to compare the performance of distance-based tree reconstruction methods. This may help explain why Neighbor-Joining performs so well, as its stochastic safety radius appears close to optimal (while its more classical safety radius is the same as many other less accurate methods).