2017/11/16 by Jie Chen, Michael L. Stein, Chen, Jie +1 · 1 citation
Environmental Science · #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Numerical Analysis (math.NA) #Remote Sensing and LiDAR Applications #Remote Sensing in Agriculture #Soil Geostatistics and Mapping
paper · pdf · doi:10.48550/arxiv.1711.05895
openalex publication_date 2017/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Gaussian random fields (GRF) are a fundamental stochastic model for spatiotemporal data analysis. An essential ingredient of GRF is the covariance function that characterizes the joint Gaussian distribution of the field. Commonly used covariance functions give rise to fully dense and unstructured covariance matrices, for which required calculations are notoriously expensive to carry out for large data. In this work, we propose a construction of covariance functions that result in matrices with a hierarchical structure. Empowered by matrix algorithms that scale linearly with the matrix dimension, the hierarchical structure is proved to be efficient for a variety of random field computations, including sampling, kriging, and likelihood evaluation. Specifically, with n scattered sites, sampling and likelihood evaluation has an O(n) cost and kriging has an O(log n) cost after preprocessing, particularly favorable for the kriging of an extremely large number of sites (e.g., predicting on more sites than observed). We demonstrate comprehensive numerical experiments to show the use of the constructed covariance functions and their appealing computation time. Numerical examples on a laptop include simulated data of size up to one million, as well as a climate data product with over two million observations.