Gaussian processes (GPs) are popular models for functions, time series, and spatial fields, but direct application of GPs is computationally infeasible for large datasets. We propose a multi-scale Vecchia (MSV) approximation of GPs for modeling and analysis of multi-scale phenomena, which are ubiquitous in geophysical and other applications. In the MSV approach, increasingly large sets of variables capture increasingly small scales of spatial variation, to obtain an accurate approximation of the spatial dependence from very large to very fine scales. For a given set of observations, the MSV approach decomposes the data into different scales, which can be visualized to obtain insights into the underlying processes. We explore properties of the MSV approximation and propose an algorithm for automatic choice of the tuning parameters. We provide comparisons to existing approaches based on simulated data and using satellite measurements of land-surface temperature.

高斯过程 (GPs) 是函数、时间序列和空间场的流行模型，但直接应用 GPs 对于大型数据集在计算上是不可行的。我们提出了 GP 的多尺度 Vecchia (MSV) 近似，用于对地球物理和其他应用中普遍存在的多尺度现象进行建模和分析。在 MSV 方法中，越来越多的变量集捕获越来越小的空间变化尺度，以获得从非常大到非常精细尺度的空间依赖性的准确近似。对于给定的一组观察结果，MSV 方法将数据分解为不同的尺度，可以将其可视化以深入了解底层过程。我们探索了 MSV 近似的特性，并提出了一种自动选择调整参数的算法。