Patchwork Kriging for Large-scale Gaussian Process Regression

This paper presents a new approach for Gaussian process (GP) regression for large datasets. The approach involves partitioning the regression input domain into multiple local regions with a different local GP model fitted in each region. Unlike existing local partitioned GP approaches, we introduce a technique for patching together the local GP models nearly seamlessly to ensure that the local GP models for two neighboring regions produce nearly the same response prediction and prediction error variance on the boundary between the two regions. This effectively solves the well-known discontinuity problem that degrades the boundary accuracy of existing local partitioned GP methods. Our main innovation is to represent the continuity conditions as additional pseudo-observations that the differences between neighboring GP responses are identically zero at an appropriately chosen set of boundary input locations. To predict the response at any input location, we simply augment the actual response observations with the pseudo-observations and apply standard GP prediction methods to the augmented data. In contrast to heuristic continuity adjustments, this has an advantage of working within a formal GP framework, so that the GP-based predictive uncertainty quantification remains valid. Our approach also inherits a sparse block-like structure for the sample covariance matrix, which results in computationally efficient closed-form expressions for the predictive mean and variance. In addition, we provide a new spatial partitioning scheme based on a recursive space partitioning along local principal component directions, which makes the proposed approach applicable for regression domains having more than two dimensions. Using three spatial datasets and three higher dimensional datasets, we investigate the numerical performance of the approach and compare it to several state-of-the-art approaches.