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：安装在卫星上的紧凑型传感设备的进步促进了带有坐标的大型时空数据集的收集。由于此类数据集通常不完整且含有噪声，构建空间场的预测曲面具有实用价值。为此，我们考虑使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很大时，卡尔曼滤波器在时间消耗上变得不切实际。为解决此问题，我们提出了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），可以在捕捉大尺度和小尺度空间变化的同时，对预测协方差矩阵进行近似。由于这种近似，我们提出的MRF-lp在时间上保持了算法中出现的某些矩阵的块稀疏结构，从而实现了算法的可扩展性。此外，我们讨论了将MRF-lp扩展到非线性和非高斯情况的可能性。针对总可降水量水汽的模拟研究和实际数据分析表明，与相关方法相比，我们提出的方法表现良好。