Bayesian Additive Regression Trees for Circular Data: A Machine Learning Framework

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Circular data arise in a wide range of scientific fields, including meteorology, medicine, biology, and neuroscience, yet existing regression methods for such data are largely restricted to parametric generalized linear models or tree-based methods that impose distributional assumptions on the circular response. In this paper, we propose a family of Bayesian Additive Regression Tree (BART) methods for regression with circular data, covering three cases: Circular–Circular BART (CCBART), where both the response and the covariates are circular; Circular–Linear BART (CLBART), where the response is circular and the covariates are linear; and Linear–Circular BART (LCBART), where the response is linear and the covariates are circular. The proposed methods adopt a projection approach, decomposing circular variables into their sine and cosine components, fitting separate BART models on these projections, and recovering circular predictions via the two-argument arctangent function. This avoids specifying a von Mises or wrapped normal likelihood directly for the circular response, though it does not avoid all distributional assumptions: BART assumes flexible Euclidean regression models, with Gaussian errors, for the projected sine and cosine components. The approach retains the full inferential power of BART, including posterior uncertainty quantification, automatic variable selection, and the ability to capture nonlinear effects and interactions without pre-specification. An extensive simulation study demonstrates that the proposed methods are highly competitive with random forest benchmarks and consistently outperform linear model benchmarks, with the clearest and most consistent advantage over random forests emerging at high noise levels and in the Linear–Circular case, where LCBART achieves up to 34% lower RMSE than projected random forests. Applications to wind direction forecasting and human motor resonance data further illustrate the practical utility of the proposed methods.