Visibility graph-based covariance functions for scalable spatial analysis in nonconvex domains
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We present a new method for constructing valid covariance functions of Gaussian processes over irregular nonconvex spatial domains such as water bodies, where the geodesic distance agrees with the Euclidean distance only for some pairs of points. Standard covariance functions based on geodesic distances are not positive definite on such domains. Using a visibility graph on the domain, we use the graphical method of "covariance selection" to propose a class of covariance functions that preserve Euclidean-based covariances between points that are connected through the domain. The proposed method preserves the partially Euclidean nature of the intrinsic geometry on the domain while maintaining validity (positive definiteness) and marginal stationarity over the entire parameter space, properties which are not always fulfilled by existing approaches to construct covariance functions on nonconvex domains. We provide useful approximations to improve computational efficiency, resulting in a scalable algorithm. We evaluate the performance of competing state-of-the-art methods using simulation studies on a contrived nonconvex domain. The method is applied to data regarding acidity levels in the Chesapeake Bay, showing its potential for ecological monitoring in real-world spatial applications on irregular domains.
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