Advances in compact sensing devices mounted on satellites have facilitated the collection of large spatio-temporal datasets with coordinates. Since such datasets are often incomplete and noisy, it is useful to create the prediction surface of a spatial field. To this end, we consider an online filtering inference by using the Kalman filter based on linear Gaussian state-space models. However, the Kalman filter is impractically time-consuming when the number of locations in spatio-temporal datasets is large. To address this problem, we propose a multi-resolution filter via linear projection (MRF-lp), a fast computation method for online filtering inference. In the MRF-lp, by carrying out a multi-resolution approximation via linear projection (MRA-lp), the forecast covariance matrix can be approximated while capturing both the large- and small-scale spatial variations. As a result of this approximation, our proposed MRF-lp preserves a block-sparse structure of some matrices appearing in the MRF-lp through time, which leads to the scalability of this algorithm. Additionally, we discuss extensions of the MRF-lp to a nonlinear and non-Gaussian case. Simulation studies and real data analysis for total precipitable water vapor demonstrate that our proposed approach performs well compared with the related methods.

翻译：卫星搭载的紧凑传感设备的进步促进了带坐标的大规模时空数据集的收集。由于此类数据集往往不完整且含有噪声，创建空间场的预测曲面具有实用价值。为此，我们考虑采用基于线性高斯状态空间模型的卡尔曼滤波器进行在线滤波推断。然而，当时空数据集中的位置数量庞大时，卡尔曼滤波器的计算耗时过大，难以实际应用。针对这一问题，我们提出一种基于线性投影的多分辨率滤波器（MRF-lp），这是一种用于在线滤波推断的快速计算方法。在MRF-lp中，通过执行基于线性投影的多分辨率近似（MRA-lp），可近似预测协方差矩阵，同时捕捉大尺度和小尺度的空间变化。通过这种近似，我们提出的MRF-lp能够保持算法中某些矩阵随时间演化的块稀疏结构，从而实现算法的可扩展性。此外，我们还讨论了MRF-lp向非线性非高斯情形的推广。针对总可降水量数据的模拟实验和真实数据分析表明，与相关方法相比，我们提出的方法表现优异。