Scalable and Robust Spatial Prediction via Multi-Resolution Ensembles of Predictive Processes

Gaussian processes provide a flexible framework for spatial prediction, but their computational cost limits applicability to large-scale data with large sample size $n$. Predictive processes (PPs), a popular low-rank approximation, mitigate this burden by projecting the original process onto a reduced set of $m\ll n$ inducing points. However, existing theory requires $m$ to grow with $n$, creating a trade-off between accuracy and computational efficiency. We address this challenge by introducing an ensemble of PPs based on spatial partitioning, and propose a novel partitioning and patching scheme with desirable properties. By generalizing the convergence results of PPs, it becomes possible to explicitly balance scalability and accuracy: increasing the number of ensemble components slows down the convergence but substantially improves computational efficiency. We further show theoretically that, despite the limited approximation accuracy of PPs with fixed $m$, they are asymptotically robust to data contamination. Motivated by this insight, we finally introduce a multi-resolution ensemble that combines PPs with fixed $m$ with multiple ensembles defined over possibly overlapping coarse to fine partitions. Simulations and large-scale geostatistical applications demonstrate that our approach delivers accurate, robust predictions with computational gains, providing a practical and broadly applicable solution for spatial prediction.