Correlated Bayesian Additive Regression Trees with Gaussian Process for Regression Analysis of Dependent Data

Bayesian Additive Regression Trees (BART) has gained widespread popularity, prompting the development of various extensions for different applications. However, limited attention has been given to analyzing dependent data. Based on a general correlated error assumption and an innovative dummy representation, we introduces a novel extension of BART, called Correlated BART (CBART), designed to handle correlated errors. By integrating CBART with a Gaussian process (GP), we propose the CBART-GP model, in which the CBART and GP components are loosely coupled, allowing them to be estimated and applied independently. CBART captures the covariate mean function E[y|x]=f(x), while the Gaussian process models the dependency structure in the response $y$. We also developed a computationally efficient approach, named two-stage analysis of variance with weighted residuals, for the estimation of CBART-GP. Simulation studies demonstrate the superiority of CBART-GP over other models, and a real-world application illustrates its practical applicability.