Tree Fitting: Topological Recognition from Ordinary Least-Squares Edge Length Estimates

Abstract

Distance methods play a central role in the field of phylogeny reconstruction, providing fast, efficient algorithms which yield reliable trees. A leading method in this field is the minimum evolution (ME) method, which selects the topology whose size is minimized when edge lengths have been selected for each topology via an ordinary least-squares (OLS) criterion. Vach and Degens (1991) have demonstrated that this OLS solution can be found by constraining edge lengths for a given topology such that average distances across each split in the tree metric equal those of the input metric. In this work we consider the vector space of tree metrics with regard to a basis generated by split average distances. This algebraic structure leads us to a conjecture regarding the edge weights of OLS tree fitting. We conjecture that if we fit a tree metric to an incorrect topology, the incorrectness will be detectable via a negative length assignment to one of the edges corresponding to a split in the test topology which is not shared by the true topology. This conjecture is proven for topologies sufficiently close to the true topology, and simulations suggest that it may be true in a more general case. We consider the durability of this topological signal in the presence of noisy sampling.