  Spatial-temporal Gaussian process regression is a popular method for
spatial-temporal data modeling. Its state-of-art implementation is based on the
state-space model realization of the spatial-temporal Gaussian process and its
corresponding Kalman filter and smoother, and has computational complexity
$\mathcal{O}(NM^3)$, where $N$ and $M$ are the number of time instants and
spatial input locations, respectively, and thus can only be applied to data
with large $N$ but relatively small $M$. In this paper, our primary goal is to
show that by exploring the Kronecker structure of the state-space model
realization of the spatial-temporal Gaussian process, it is possible to further
reduce the computational complexity to $\mathcal{O}(M^3+NM^2)$ and thus the
proposed implementation can be applied to data with large $N$ and moderately
large $M$. The proposed implementation is illustrated over applications in
weather data prediction and spatially-distributed system identification. Our
secondary goal is to design a kernel for both the Colorado precipitation data
and the GHCN temperature data, such that while having more efficient
implementation, better prediction performance can also be achieved than the
state-of-art result.
