  In past decades, Gaussian processes has been widely applied in studying trait
evolution using phylogenetic comparative analysis. In particular, two members
of Gaussian processes: Brownian motion and Ornstein-Uhlenbeck process, have
been frequently used to describe continuous trait evolution. Under the
assumption of adaptive evolution, several models have been created around
Ornstein-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 rate
of evolution $\tau^y_t$ of trait could adopt a pertinent process, in this work
we extend models of adaptive evolution by considering the rate of evolution
$\tau_t^y$ following the Cox-Ingersoll-Ross (CIR) process. We provide a
heuristic Monte Carlo simulation scheme to simulate trait along the phylogeny
as a structure of dependence among species. We add a framework to incorporate
multiple regression with interaction between optimum of the trait and its
potential predictors. Since the likelihood function for our models are
intractable, we propose the use of Approximate Bayesian Computation (ABC) for
parameter estimation and inference. Simulation as well as empirical study using
the proposed models are also performed and carried out to validate our models
and for practical applications.
