Gaussian processes on phylogenies: a kernel framework for building, checking and comparing models of trait evolution
Abstract
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.
Related
- Evolution of the axial skeleton of early dinosaurs and the role of meristic and homeotic processes
- The relative psychometric function: A general analysis framework for relating psychological processes.
- Phylogenies as graphs: structured neural networks improve host origin predictions from paramyxovirus sequences
- Deep learning for mass extinction detection on fossilized phylogenies: power, limitations and lessons for simulation-based birth–death inference
- An innovative framework based on GMV-ALFF multi-modal MRI and Gaussian data augmentation for psychiatric disorder classification
BibTeX
@misc{puschel2026gaussian,
title = {{Gaussian processes on phylogenies: a kernel framework for building, checking and comparing models of trait evolution}},
author = {Thomas A Püschel},
howpublished = {EcoEvoRxiv},
year = {2026},
month = oct,
doi = {10.32942/x2d99v},
url = {https://ecoevorxiv.org/repository/view/15619/},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian processes on phylogenies: a kernel framework for building, checking and comparing models of trait evolution
AU - Thomas A Püschel
PY - 2026
DA - 2026/10/22
PB - EcoEvoRxiv
DO - 10.32942/x2d99v
UR - https://ecoevorxiv.org/repository/view/15619/
AB - 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.
ER -