In geostatistics, traditional spatial models often rely on the Gaussian Process (GP) to fit stationary covariances to data. It is well known that this approach becomes computationally infeasible when dealing with large data volumes, necessitating the use of approximate methods. A powerful class of methods approximate the GP as a sum of basis functions with random coefficients. Although this technique offers computational efficiency, it does not inherently guarantee a stationary covariance. To mitigate this issue, the basis functions can be "normalized" to maintain a constant marginal variance, avoiding unwanted artifacts and edge effects. This allows for the fitting of nearly stationary models to large, potentially non-stationary datasets, providing a rigorous base to extend to more complex problems. Unfortunately, the process of normalizing these basis functions is computationally demanding. To address this, we introduce two fast and accurate algorithms to the normalization step, allowing for efficient prediction on fine grids. The practical value of these algorithms is showcased in the context of a spatial analysis on a large dataset, where significant computational speedups are achieved. While implementation and testing are done specifically within the LatticeKrig framework, these algorithms can be adapted to other basis function methods operating on regular grids.

翻译：在地统计学中，传统空间模型通常依赖高斯过程（GP）将平稳协方差函数拟合至数据。众所周知，当处理大规模数据时，该方法在计算上变得不可行，因此必须采用近似方法。一类有效的方法是将GP近似为带随机系数的基函数之和。尽管该技术提供了计算效率，但其本身并不能保证协方差的平稳性。为缓解此问题，可对基函数进行“归一化”以保持恒定的边缘方差，从而避免非预期的伪影与边界效应。这使得能够对大规模（可能非平稳）数据集拟合近似平稳的模型，为扩展至更复杂问题提供了严谨基础。然而，归一化这些基函数的过程计算量巨大。为此，我们针对归一化步骤提出了两种快速且精确的算法，从而支持在精细网格上进行高效预测。这些算法的实用价值通过一项大规模数据集的空间分析案例得以展示，其中实现了显著的计算加速。虽然具体实现与测试在LatticeKrig框架内完成，但这些算法可适配于其他在规则网格上运行的基函数方法。