A multilevel Ornstein–Uhlenbeck process with individual- and variable-specific estimates as random effects

Abstract

In the present study, we extend a stochastic differential equation (SDE) model, the Ornstein–Uhlenbeck (OU) process, to the simultaneous analysis of time series of multiple variables by means of random effects for individuals and variables using a Bayesian framework. This SDE model is a stationary Gauss-Markov process that varies over time around its mean. Our extension allows us to estimate the variability of different parameters of the process, such as the mean (μ) or the drift parameter (φ), across individuals and variables of the system by means of marginalized posterior distributions. We illustrate the estimations and the interpretability of the parameters of this multilevel OU process in an empirical study of affect dynamics where multiple individuals were measured on different variables at multiple time points. We also conducted a simulation study to evaluate whether the model can recover the population parameters generating the OU process. Our results support the use of this model to obtain both the general parameters (common to all individuals and variables) and the variable-specific point estimates (random effects). We conclude that this multilevel OU process with individual- and variable-specific estimates as random effects can be a useful approach to analyse time series for multiple variables simultaneously.