Abstract: Gaussian processes are popular and flexible models for spatial, temporal, and functional data, but they are omputationally infeasible for large data sets. We discuss Gaussian-process approximations that use basis functions at multiple resolutions to achieve fast inference and that can (approximately) represent any spatial covariance structure. We consider two special cases of this multi-resolution approximation framework, namely a taper version and a domain-partitioning (block) version. We describe theoretical properties and inference procedures, and study the computational complexity of the methods. Numerical comparisons and an application to satellite data are also provided.

Key words and phrases: Basis functions, Gaussian process, kriging, predictive process, satellite data, sparsity.