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In past decades, Gaussian processes has been widely applied in studying traitevolution using phylogenetic comparative analysis. In particular, two membersof Gaussian processes: Brownian motion and Ornstein-Uhlenbeck process, havebeen frequently used to describe continuous trait evolution. Under theassumption of adaptive evolution, several models have been created aroundOrnstein-Uhlenbeck process where the optimum $\theta^y_t$ of a single trait$y_t$ is influenced with predictor $x_t$. Since in general the dynamics of rateof evolution $\tau^y_t$ of trait could adopt a pertinent process, in this workwe extend models of adaptive evolution by considering the rate of evolution$\tau_t^y$ following the Cox-Ingersoll-Ross (CIR) process. We provide aheuristic Monte Carlo simulation scheme to simulate trait along the phylogenyas a structure of dependence among species. We add a framework to incorporatemultiple regression with interaction between optimum of the trait and itspotential predictors. Since the likelihood function for our models areintractable, we propose the use of Approximate Bayesian Computation (ABC) forparameter estimation and inference. Simulation as well as empirical study usingthe proposed models are also performed and carried out to validate our modelsand for practical applications.

Statistical modeling for adaptive trait evolution in randomly evolving environment