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.

翻译：我们提出了一种新方法，用于构建高斯过程在非凸不规则空间域（如水体）上的有效协方差函数，其中测地距离仅在部分点对间与欧氏距离一致。基于测地距离的标准协方差函数在此类域上非正定。利用域上的可见性图，我们采用"协方差选择"的图论方法，提出了一类协方差函数，该函数保留域内连通点之间的欧氏协方差。所提方法既保持了域内固有几何的部分欧氏性质，又确保了整个参数空间上的有效性（正定性）和边际平稳性——这些性质是现有非凸域协方差函数构建方法未必满足的。我们提供了有效近似以提高计算效率，从而得出可扩展算法。通过在人造非凸域上的仿真研究，评估了竞争性前沿方法的性能。该方法被应用于切萨皮克湾酸度水平数据，展示了其在真实世界不规则空间域生态监测中的应用潜力。