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

Thomas A Püschel

Phylogenetic comparative methods for continuous traits, such as phylogenetic generalised least squares, Brownian motion and Ornstein–Uhlenbeck models, Pagel's tree transformations, early-burst and multi-rate models, phylogenetic mixed models and ancestral-state reconstruction, 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 the present study, I use it to build, check and compare models of trait evolution. Most classical models share a single branch-wise form of the kernel, 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. A generic implementation in Stan reproduced standard software to numerical precision on 99.6% of simulated data sets, on ultrametric trees, trees with fossil tips and trees with polytomies. To check models, I collect the conditions under which kernels defined on patristic distances are valid. Any non-increasing function is valid 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 and Matérn kernels can fail. I also provide a numerical test of whether two kernels imply the same distribution for the tips, which recovers the known equivalence between single-optimum Ornstein–Uhlenbeck and accelerating-rate models on ultrametric trees. To build models, I write four extensions as kernels (i.e. autocorrelated rates, smooth rate-through-time functions, joint phylogenetic and spatial structure, and phylogenetically varying regression slopes), all of which have close precedents in the literature, and I assess their bias, interval coverage, power and false-positive rates 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, where kernel extensions were compared with simpler explanations (e.g. heavy-tailed residuals and a curved allometric relationship) and validated by leaving out whole orders. All methods are implemented in the R package phylogp. Overall, treating comparative methods as Gaussian processes offers a common and extensible basis both for developing new models of trait evolution and for testing them.