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.

翻译：本文提出了一种新方法，用于构建高斯过程的有效协方差函数，以应用于水体等不规则非凸域的空间分析。基于测地线距离的标准协方差函数在此类域上无法保证正定性，而现有的非欧几里得方法未能兼顾这些域的部分欧几里得特性——即对于某些点对，其测地距离与欧几里得距离一致。通过在域上构建可见性图，我们提出了一类协方差函数，该函数能保持域内连通点对间基于欧几里得度量的协方差关系，同时通过条件独立关系融入域的非凸几何特征。我们证明，所提方法在保持协方差函数在整个参数空间上的有效性（正定性）与边缘平稳性的同时，保留了域内禀几何的部分欧几里得特性，而这些特性在现有非凸域协方差函数构建方法中往往无法同时满足。我们提供了有效的近似策略以提升计算效率，从而形成可扩展的算法。通过在合成非凸域上的模拟研究，我们将本方法与当前主流先进方法进行了性能对比。该方法已应用于切萨皮克湾酸度数据，展现了其在真实世界不规则域空间分析中用于生态监测的潜力。