Understanding the processes that give rise to quantitative measurements associated with molecular sequence data remains an important issue in statistical phylogenetics. Examples of such measurements include geographic coordinates in the context of phylogeography and phenotypic traits in the context of comparative studies. A popular approach is to model the evolution of continuously varying traits as a Brownian diffusion process acting on a phylogenetic tree. However, standard Brownian diffusion is quite restrictive and may not accurately characterize certain trait evolutionary processes. Here, we relax one of the major restrictions of standard Brownian diffusion by incorporating a nontrivial estimable mean into the process. We introduce a relaxed directional random walk (RDRW) model for the evolution of multivariate continuously varying traits along a phylogenetic tree. Notably, the RDRW model accommodates branch-specific variation of directional trends while preserving model identifiability. Furthermore, our development of a computationally efficient dynamic programming approach to compute the data likelihood enables scaling of our method to large data sets frequently encountered in phylogenetic comparative studies and viral evolution. We implement the RDRW model in a Bayesian inference framework to simultaneously reconstruct the evolutionary histories of molecular sequence data and associated multivariate continuous trait data, and provide tools to visualize evolutionary reconstructions. We demonstrate the performance of our model on synthetic data, and we illustrate its utility in two viral examples. First, we examine the spatiotemporal spread of HIV-1 in central Africa and show that the RDRW model uncovers a clearer, more detailed picture of the dynamics of viral dispersal than standard Brownian diffusion. Second, we study antigenic evolution in the context of HIV-1 resistance to three broadly neutralizing antibodies. Our analysis reveals evidence of a continuous drift at the HIV-1 population level towards enhanced resistance to neutralization by the VRC01 monoclonal antibody over the course of the epidemic. [Brownian Motion; Diffusion Processes; Phylodynamics; Phylogenetics; Phylogeography; Trait Evolution.].
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