Non-linear phylogenetic regression using regularised kernels

Phylogenetic regression is a type of generalised least squares (GLS) method that incorporates a modelled covariance matrix based on the evolutionary relationships between species (i.e. phylogenetic relationships). While this method has found widespread use in hypothesis testing via phylogenetic comparative methods, such as phylogenetic ANOVA, its ability to account for non-linear relationships has received little attention. To address this, here we implement a phylogenetic Kernel Ridge Regression (phyloKRR) method that utilises GLS in a high-dimensional feature space, employing linear combinations of phylogenetically weighted data to account for non-linearity. We analysed two biological datasets using the Radial Basis Function and linear kernel function. The first dataset contained morphometric data, while the second dataset comprised discrete trait data and diversification rates as response variable. Hyperparameter tuning of the model was achieved through cross-validation rounds in the training set. In the tested biological datasets, phyloKRR reduced the error rate (as measured by RMSE) by around 20% compared to linear-based regression when data did not exhibit linear relationships. In simulated datasets, the error rate decreased almost exponentially with the level of non-linearity. These results show that introducing kernels into phylogenetic regression analysis presents a novel and promising tool for complementing phylogenetic comparative methods. We have integrated this method into Python package named phyloKRR, which is freely available at: .