A ‘Stochastic Safety Radius’ for Distance-Based Tree Reconstruction

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 species. 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).

Appendix: Proof of (2)

Substituting $t=x+u, u \ge 0$ in ${\mathbb P}(N(0,1)>x) = \int _x^\infty \frac{1}{\sqrt{2\pi }} e^{-t^2/2} dt$ gives:

$$\begin{aligned} {\mathbb P}(N(0,1)>x) = e^{-x^2/2}\int _0^\infty \frac{1}{\sqrt{2\pi }} e^{-xu}e^{-u^2/2} du < e^{-x^2/2}\int _0^\infty \frac{1}{\sqrt{2\pi }} e^{-u^2/2} du, \end{aligned}$$

where the second inequality is from $e^{-xu}< 1$ for all $x,u>0$. Since the last term on the right is $\frac{1}{2}$, we get the inequality in (2). Turning to the asymptotic relationship, consider:

$$\begin{aligned} \lim _{x \rightarrow \infty } \frac{\frac{1}{\sqrt{2\pi }}\int _x^\infty e^{-t^2/2} dt}{\frac{1}{x\sqrt{2\pi }} e^{-x^2/2}}. \end{aligned}$$

(13)

Since the numerator and denominator limits are both zero, we can apply L’Hôpital’s rule. Straightforward calculus (using the fundamental theorem of calculus for the numerator) establishes that the limit in (13) equals 1. $\square$