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 simulations 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）上的合法协方差函数，其中测地距离仅对部分点对与欧氏距离一致。基于测地距离的标准协方差函数在此类域上并非正定。通过利用域内的可视性图，我们采用“协方差选择”的图方法，提出了一类协方差函数，该函数保留了域内连通点之间的基于欧氏距离的协方差。所提方法在保持域内固有几何的部分欧氏性质的同时，确保了整个参数空间上的合法性（正定性）和边际平稳性，而现有非凸域协方差函数构建方法并非总能满足这些性质。我们提供了有助于提高计算效率的近似方法，从而实现了可扩展的算法。通过在一个虚构的非凸域上进行的仿真研究，我们评估了与现有最先进方法的性能对比。该方法被应用于切萨皮克湾酸度水平数据，展示了其在非规则域实际空间应用中用于生态监测的潜力。