Sparse Pseudo-input Local Kriging for Large Non-stationary Spatial Datasets with Exogenous Variables

We study large-scale spatial systems that contain exogenous variables, e.g. environmental factors that are significant predictors in spatial processes. Building predictive models for such processes involve two major challenges. First, the spatial processes are highly non-stationary and characterized by a heterogeneous covariance structure primarily due to the presence of exogenous variables. Second, it is inefficient to apply full Kriging because of the large numbers of observations present. The new theorems proposed in this paper form the basis for a new partitioning policy and a method, which we call Sparse Pseudo-input Local Kriging (SPLK). The proposed method handles heterogeneity in covariance and computational complexity by utilizing hyperplanes to partition a domain into smaller subdomains and then applying a sparse approximation of the full Kriging to each subdomain. We also develop a procedure to find the optimal hyperplanes. We impose continuity constraints on the boundaries of the neighboring subdomains to alleviate the problem of discontinuity of the global predictor. Numerical experiments demonstrate that SPLK outperforms, or is comparable to, the algorithms commonly applied to spatial datasets.