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Advances in compact sensing devices mounted on satellites have facilitatedthe collection of large spatio-temporal datasets with coordinates. Since suchdatasets are often incomplete and noisy, it is useful to create the predictionsurface of a spatial field. To this end, we consider an online filteringinference by using the Kalman filter based on linear Gaussian state-spacemodels. However, the Kalman filter is impractically time-consuming when thenumber of locations in spatio-temporal datasets is large. To address thisproblem, we propose a multi-resolution filter via linear projection (MRF-lp), afast computation method for online filtering inference. In the MRF-lp, bycarrying out a multi-resolution approximation via linear projection (MRA-lp),the forecast covariance matrix can be approximated while capturing both thelarge- and small-scale spatial variations. As a result of this approximation,our proposed MRF-lp preserves a block-sparse structure of some matricesappearing in the MRF-lp through time, which leads to the scalability of thisalgorithm. Additionally, we discuss extensions of the MRF-lp to a nonlinear andnon-Gaussian case. Simulation studies and real data analysis for totalprecipitable water vapor demonstrate that our proposed approach performs wellcompared with the related methods.

Multi-resolution filters via linear projection for large spatio-temporal datasets