Equivalent kriging

Abstract

Most modern spatially indexed datasets are very large, with sizes commonly ranging from tens of thousands to millions of locations. Spatial analysis often focuses on spatial smoothing using the geostatistical technique known as kriging. Kriging requires covariance matrix computations whose complexity scales with the cube of the number of spatial locations, making analysis infeasible or impossible with large datasets. We introduce an approach to kriging in the presence of large datasets called equivalent kriging, which relies on approximating the kriging weight function using an equivalent kernel, requiring presence of a nontrivial nugget effect. Resulting kriging calculations are extremely fast and feasible in the presence of massive spatial datasets. We derive closed form kriging approximations for multiresolution classes of spatial processes, as well as under any stationary model, including popular choices such as the Matérn. The theoretical justification for equivalent kriging also leads to a correction term for irregularly spaced observations that also reduces edge effects near the domain boundary. For large sample sizes, equivalent kriging is shown to outperform covariance tapering in an example. Equivalent kriging is additionally illustrated on multiple simulated datasets, and a monthly average precipitation dataset whose size prohibits traditional geostatistical approaches.