Random fields on the hypertorus: Covariance modeling and applications

This article gives a comprehensive theoretical framework to the modeling, inference, and applications of Gaussian random fields using what we term the hypertorus as an index set. The hypertorus is obtained through a product of hyperspheres. We envision the following as appropriate settings for random fields on the hypertorus: continuous‐time data with multiple sources of seasonality, directional data with seasonality or over the globe, and global spatiotemporal data with temporal seasonality. We propose modeling strategies for such data through covariance structures over the hypertorus. We develop various families of covariance functions over the hypertorus and discuss how to construct random fields using these covariance functions. We show the utility of our findings on three datasets. Our first example is a dataset of ozone concentrations from Mexico City that exhibits multiple sources of seasonality. Our second dataset is a wind speed dataset, where the data show daily seasonality and are indexed by wind direction. Our third illustration considers a global space‐time dataset of cloud coverage, demonstrating strong seasonality. In all analyses, we compare the predictive performance of random fields specified through various covariance structures and examine the results of the best predictive model.