Statistical inference for spatial processes from partially realized or scattered data has seen voluminous developments in diverse areas ranging from environmental sciences to business and economics. Inference on the associated rates of change has seen some recent developments. The literature has been restricted to Euclidean domains, where inference is sought on directional derivatives, rates along a chosen direction of interest, at arbitrary locations. Inference for higher order rates, particularly directional curvature has also proved useful in these settings. Modern spatial data often arise from non-Euclidean domains. This manuscript particularly considers spatial processes defined over compact Riemannian manifolds. We develop a comprehensive inferential framework for spatial rates of change for such processes over vector fields. In doing so, we formalize smoothness of process realizations and construct differential processes -- the derivative and curvature processes. We derive conditions for kernels that ensure the existence of these processes and establish validity of the joint multivariate process consisting of the ``parent'' Gaussian process (GP) over the manifold and the associated differential processes. Predictive inference on these rates is devised conditioned on the realized process over the manifold. Manifolds arise as polyhedral meshes in practice. The success of our simulation experiments for assessing derivatives for processes observed over such meshes validate our theoretical findings. By enhancing our understanding of GPs on manifolds, this manuscript unlocks a variety of potential applications in machine learning and statistics where GPs have seen wide usage. We propose a fully model-based approach to inference on the differential processes arising from a spatial process from partially observed or realized data across scattered location on a manifold.

翻译：从部分实现或散点数据中推断空间过程的统计方法已在环境科学到商业经济等众多领域得到广泛发展。关于相关变化率的推断近年来也取得了一些进展。现有文献主要局限于欧几里得域，旨在对任意位置处沿特定感兴趣方向的方向导数及变化率进行推断。在这些场景中，高阶变化率（特别是方向曲率）的推断也被证明具有实用价值。现代空间数据常产生于非欧几里得域。本文特别关注定义在紧致黎曼流形上的空间过程。我们为此类过程在向量场上的空间变化率构建了完整的推断框架。在此过程中，我们形式化了过程实现的光滑性，并构造了微分过程——导数过程与曲率过程。我们推导了确保这些过程存在的核函数条件，并建立了由流形上的“父”高斯过程（GP）及其关联微分过程组成的联合多元过程的有效性。基于流形上已实现过程的条件，我们设计了这些变化率的预测推断方法。在实际应用中，流形常以多面体网格形式呈现。我们在该类网格上观测过程并评估导数的模拟实验成功验证了理论结果。通过深化对流形上高斯过程的理解，本文为机器学习和统计学中广泛使用高斯过程的诸多潜在应用开辟了新途径。我们提出了一种完全基于模型的推断方法，用于处理流形上散点位置处部分观测或已实现数据所生成空间过程的微分过程推断。