Geostatistics is a branch of statistics concerned with stochastic processes over continuous domains, with Gaussian processes (GPs) providing a flexible and principled modelling framework. However, the high computational cost of simulating or computing likelihoods with GPs limits their scalability to large datasets. This paper introduces the piecewise continuous Gaussian process (PCGP), a new process that retains the rich probabilistic structure of traditional GPs while offering substantial computational efficiency. As will be shown and discussed, existing scalable approaches that define stochastic processes on continuous domains -- such as the nearest-neighbour GP (NNGP) and the radial-neighbour GP (RNGP) -- rely on conditional independence structures that effectively constrain the measurable space on which the processes are defined, which may induce undesirable probabilistic behaviour and compromise their practical applicability, particularly in complex latent GP models. The PCGP mitigates these limitations and provides a theoretically grounded and computationally efficient alternative, as demonstrated through numerical illustrations.

翻译：地质统计学是统计学分支，主要研究连续域上的随机过程，其中高斯过程（GP）提供了灵活且原则性的建模框架。然而，高斯过程模拟或计算似然的高昂计算成本限制了其在大型数据集上的可扩展性。本文提出分段连续高斯过程（PCGP）这一新方法，在保留传统高斯过程丰富概率结构的同时，显著提升计算效率。后续将论证并讨论：现有定义在连续域上的可扩展方法——如邻近高斯过程（NNGP）和径向邻近高斯过程（RNGP）——所依赖的条件独立性结构实际上约束了过程的定义可测空间，这可能引发不良概率行为并损害其实践适用性，尤其在复杂潜在高斯过程模型中。通过数值实验证明，PCGP方法缓解了上述局限性，为实践提供了具备理论基础且计算高效的新方案。