Gaussian processes on phylogenies: a kernel framework for building, checking and comparing models of trait evolution

Thomas A Püschel

1. Phylogenetic comparative methods for continuous traits (phylogenetic generalised least squares, Brownian motion and Ornstein–Uhlenbeck models, Pagel's transformations, early-burst and multi-rate models, phylogenetic mixed models) can all be regarded as Gaussian processes on a phylogeny that differ only in their covariance function, or kernel. This link has been noted before, but it has not been developed into a working framework in which models are built, checked and compared in the same way.

2. In the present study, I write the classical models as kernels in one notation and state where each representation holds. Most of them share a branch-wise form, which makes it possible to compute likelihoods, leave-one-out and leave-clade-out predictive densities, and predictions at tips and internal nodes in a time that grows linearly with the number of species. I also collect the conditions under which kernels defined on patristic distances are valid covariances, provide a numerical test of whether two kernels imply the same distribution for the tips, and write four extensions as kernels (i.e. autocorrelated rates, smooth rate-through-time functions, joint phylogenetic and spatial structure, and phylogenetically varying slopes). I assessed implementation equivalence against standard software and the statistical behaviour of the extensions by simulation on pure-birth trees and on subtrees of an empirical phylogeny. As an empirical example, I applied the framework to brain–body allometry in 1,349 mammal species. All methods are implemented in the R package `phylogp`.

3. A generic implementation in Stan reproduced standard software to numerical precision in 698 of 700 simulated data sets, on ultrametric trees, trees with fossil tips and trees with polytomies. The remaining two were differences between optimisers at a parameter boundary. Any non-increasing function of patristic distance is a valid kernel on ultrametric trees. However, when fossil tips are included only completely monotone functions (e.g. the exponential) are guaranteed to be valid, and the squared-exponential kernel failed at 73% of the length-scales examined. The extended kernels were never preferred over simpler models when the process was absent. In addition, power was limited for weak effects and abrupt rate shifts were shrunk. In the mammal example, most of the apparent predictive gain of the extended kernels came from heavy-tailed residuals and curvature, and leaving out whole orders reversed the advantage of varying slopes.

4. Overall, treating comparative methods as Gaussian processes offers a common and extensible basis both for developing new models of trait evolution and for checking them before they are used. Therefore, predictive comparisons should be made against simpler explanations that leave the same statistical signature.