Visibility graph-based covariance functions for scalable spatial analysis in non-convex partially Euclidean domains

Area of Science:

• Spatial Statistics
• Geostatistics
• Environmental Science

Background:

• Standard Gaussian process covariance functions struggle with irregular, non-convex domains, lacking positive definiteness.
• Existing non-Euclidean methods overlook the partially Euclidean nature of some irregular domains.

Purpose of the Study:

• To propose a new method for constructing valid covariance functions for Gaussian processes in irregular, non-convex domains.
• To address limitations of existing methods in preserving Euclidean properties and ensuring positive definiteness.

Main Methods:

• Utilized a visibility graph to construct covariance functions, connecting points within the domain.
• Incorporated conditional independence relationships to account for non-convex geometry.
• Developed approximations for computational efficiency, creating a scalable algorithm.

Main Results:

• The proposed method ensures valid (positive definite) and marginally stationary covariance functions.
• Preserves the partially Euclidean nature of the domain's intrinsic geometry.
• Demonstrated superior performance compared to state-of-the-art methods in simulation studies.

Conclusions:

• The new method offers a robust solution for spatial analysis in complex, irregular domains.
• Applicable to real-world ecological monitoring, such as acidity level analysis in the Chesapeake Bay.
• Provides a scalable and computationally efficient approach for geostatistical modeling.