On the Matrix Condition of Phylogenetic Tree

Abstract

Phylogenetic comparative analyses use trees of evolutionary relationships between species to understand their evolution and ecology. A phylogenetic tree of n taxa can be algebraically transformed into an n by n squared symmetric phylogenetic covariance matrix C where each element $c_{\mathrm{ij}}$ in C represents the affinity between extant species i and extant species j. This matrix C is used internally in several comparative methods: for example, it is often inverted to compute the likelihood of the data under a model. However, if the matrix is ill-conditioned (ie, if $\kappa$ , defined by the ratio of the maximum eigenvalue of C to the minimum eigenvalue of C , is too high), this inversion may not be stable, and thus neither will be the calculation of the likelihood or parameter estimates that are based on optimizing the likelihood. We investigate this potential issue and propose several methods to attempt to remedy this issue.

Keywords: Brownian motion; condition number; covariance matrix inversion; phylogenetic comparative analysis; phylogenetic tree.