Towards a Complete Picture of Covariance Functions on Spheres Cross Time
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With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides a characterization theorem for covariance functions of random fields defined over d-dimensional spheres cross time. The result characterizes the relationship between covariance functions for spheres over time and space-time covariance functions defined over Euclidean spaces. We then show that the Gneiting class of space-time covariance functions gneiting2002 can be extended to spheres cross time by replacing the squared Euclidean distance with the great circle distance. Additionally, we provide a new class of covariance functions using our characterization theorem, giving an example of a class correspondence between covariance functions over Euclidean spaces with compact support and covariance functions over spheres cross time. We discuss modeling details using nearest-neighbor Gaussian processes in a Bayesian framework for our extension of the Gneiting class. In this context, we illustrate the value of our proposed classes by comparing them to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on a simulated dataset and two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. In our simulation study, we establish that covariance parameters from a generative model from our class can be identified in model fitting and that predictive performance is as good or better than competing covariance models. In our real data examples, we show that our covariance class has better predictive performance than competing models, and we discuss results in the context of the climate processes that we model.
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