Bayesian hierarchical Ornstein-Uhlenbeck models for intervention designs

The Ornstein-Uhlenbeck (OU) model represents time series data as mean-reverting stochastic processes that gravitate towards a particular level. The OU model is widely used in fields such as finance, physics, and biology to model the dynamics of non-linear time series data. The model has three main parameters, namely the attractor, the elasticity, and the volatility, which are interpreted as the steady-state level of the variable, the speed of reversion to the mean, and the intensity of random fluctuations around the mean, respectively. Hierarchical Bayesian implementations of the OU model allow for flexible and robust data analysis by incorporating population-level parameters and individual-level heterogeneity. It also allows flexibility in the structure of the model, so that we can include time-varying parameters, latent class indicators, and relevant prior information. We apply a Bayesian hierarchical OU model to data from a mobile health intervention design aimed at promoting psychological well-being in college students. The model allows us to estimate the effectiveness of the intervention on psychological well-being over time, the persistence of the effect after the intervention, and to identify individual-level features of the response to the intervention.