Permutation and Grouping Methods for Sharpening Gaussian Process Approximations

Vecchia’s approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the article includes a systematic study of how ordering affects the accuracy of Vecchia’s approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this article is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback–Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented.针对高斯过程（Gaussian Process）参数的Vecchia近似似然，其性能依赖于观测值的排序方式，这一点曾被学界指出为该方法的固有缺陷。本文提出了一种差异化的研究视角：可通过调整观测值的排序方式来优化近似效果的精度。诚然，本文第一部分系统探究了排序方式对Vecchia近似精度的影响机制。研究得到了令人意外的结论：随机排序所得到的近似效果，其精度远优于默认的基于坐标的排序方式。本文还描述并通过数值分析验证了多种额外的排序方案，其中不乏性能超越随机排序的排序策略。本文的第二项核心贡献，是提出了一种全新的自动分组方法，用于对近似计算的各分量进行分组处理。该分组方法可在提升近似精度的同时，有效降低计算负载。在常规应用场景下，结合重排序与分组策略后，相较于采用默认排序的未分组近似方法，本文方法可将与目标模型间的库尔贝克-莱布勒散度（Kullback–Leibler Divergence）降低60倍以上。本文的所有结论均通过理论推导与数值实验予以验证，同时还与其他近似方法（包括锥化协方差（Tapered Covariances）与随机偏微分方程（Stochastic Partial Differential Equations））开展了对比分析。本文还给出了详细的计算实现细节，涵盖该近似方法在预测与条件模拟中的应用方式。最后展示了该方法在时空卫星数据场景下的实际应用案例。