A Non-Parametric Framework for Correlation Functions on Product Metric Spaces

We propose a non-parametric framework for analysing data defined over products of metric spaces, a versatile class encountered in various fields. This framework accommodates non-stationarity and seasonality and is applicable to both local and global domains, such as the Earth's surface, as well as domains evolving over linear time or time embedded in more complex geometries. Our focus is on defining random correlation functions over these spaces, which we achieve through the introduction of a symplectic double random sequence, developed using a novel constructive approach termed compositional stick-breaking. We explore the distributional properties of the resulting random correlation functions, particularly when used as non-parametric priors for unknown correlation structures in a Bayesian framework. Because this modelling strategy relies on infinite orthogonal expansions, and truncations are used to ensure computational feasibility, we analyse the impact of these truncations by deriving probabilistic upper bounds for the approximations they introduce. Finally, we demonstrate the method's performance through applications to synthetic data and real wind speed data on the hypertorus.