Visibility graph-based covariance functions for scalable spatial analysis in non-convex domains
arXiv (Cornell University)(2023)
摘要
We present a new method for constructing valid covariance functions of
Gaussian processes for spatial analysis in irregular, non-convex domains such
as bodies of water. Standard covariance functions based on geodesic distances
are not guaranteed to be positive definite on such domains, while existing
non-Euclidean approaches fail to respect the partially Euclidean nature of
these domains where the geodesic distance agrees with the Euclidean distances
for some pairs of points. Using a visibility graph on the domain, we propose a
class of covariance functions that preserve Euclidean-based covariances between
points that are connected in the domain while incorporating the non-convex
geometry of the domain via conditional independence relationships. We show that
the proposed method preserves the partially Euclidean nature of the intrinsic
geometry on the domain while maintaining validity (positive definiteness) and
marginal stationarity of the covariance function over the entire parameter
space, properties which are not always fulfilled by existing approaches to
construct covariance functions on non-convex domains. We provide useful
approximations to improve computational efficiency, resulting in a scalable
algorithm. We compare the performance of our method with those of competing
state-of-the-art methods using simulation studies on synthetic non-convex
domains. 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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关键词
spatial analysis,nonconvex domains,covariance functions,graph-based
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