Multi-resolution filters for massive spatio-temporal data

Spatio-temporal datasets are rapidly growing in size. For example, environmental variables are measured with increasing resolution by increasing numbers of automated sensors mounted on satellites and aircraft. Using such data, which are typically noisy and incomplete, the goal is to obtain complete maps of the spatio-temporal process, together with uncertainty quantification. We focus here on real-time filtering inference in linear Gaussian state-space models. At each time point, the state is a spatial field evaluated on a very large spatial grid, making exact inference using the Kalman filter computationally infeasible. Instead, we propose a multi-resolution filter (MRF), a highly scalable and fully probabilistic filtering method that resolves spatial features at all scales. We prove that the MRF matrices exhibit a particular block-sparse multi-resolution structure that is preserved under filtering operations through time. We describe connections to existing methods, including hierarchical matrices from numerical mathematics. We also discuss inference on time-varying parameters using an approximate Rao-Blackwellized particle filter, in which the integrated likelihood is computed using the MRF. Using a simulation study and a real satellite-data application, we show that the MRF strongly outperforms competing approaches. Supplementary materials include Python code for reproducing the simulations, some detailed properties of the MRF and auxiliary theoretical results.

时空（spatio-temporal）数据集的规模正快速增长。例如，搭载于卫星与航空器的自动化传感器数量持续增加、分辨率不断提升，正用于采集环境变量数据。针对这类通常存在噪声且不完整的数据，研究目标是获取完整的时空过程全域场图，并同步完成不确定性量化（uncertainty quantification）。本文聚焦于线性高斯状态空间模型（linear Gaussian state-space models）中的实时滤波推理任务。在每个时间节点，系统状态为在超大型空间网格上离散取值的空间场域，这使得借助卡尔曼滤波（Kalman filter）进行精确推理在计算上不可行。为此，本文提出了多分辨率滤波（multi-resolution filter，简称MRF）——一种具备高可扩展性且完全概率化的滤波方法，可对全尺度空间特征进行解析。本文证明，MRF矩阵具备特定的分块稀疏多分辨率结构，且该结构在时序滤波运算中得以保持。本文还阐述了该方法与现有技术的关联，包括数值数学领域的分层矩阵。此外，本文探讨了基于近似拉奥-布莱克韦尔化粒子滤波（Rao-Blackwellized particle filter）的时变参数推理任务，该方法通过MRF计算积分似然（integrated likelihood）。通过仿真实验与真实卫星数据应用案例，本文证明MRF的性能显著优于同类竞争方法。补充材料包含用于复现仿真实验的Python代码、MRF的部分详细特性以及辅助理论推导结果。