  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.
