  As a regression technique in spatial statistics, the spatiotemporally varying
coefficient model (STVC) is an important tool for discovering nonstationary and
interpretable response-covariate associations over both space and time.
However, it is difficult to apply STVC for large-scale spatiotemporal analyses
due to its high computational cost. To address this challenge, we summarize the
spatiotemporally varying coefficients using a third-order tensor structure and
propose to reformulate the spatiotemporally varying coefficient model as a
special low-rank tensor regression problem. The low-rank decomposition can
effectively model the global patterns of large data sets with a substantially
reduced number of parameters. To further incorporate the local spatiotemporal
dependencies, we use Gaussian process (GP) priors on the spatial and temporal
factor matrices. We refer to the overall framework as Bayesian Kernelized
Tensor Regression (BKTR), and kernelized tensor factorization can be considered
a new and scalable approach to modeling multivariate spatiotemporal processes
with a low-rank covariance structure. For model inference, we develop an
efficient Markov chain Monte Carlo (MCMC) algorithm, which uses Gibbs sampling
to update factor matrices and slice sampling to update kernel hyperparameters.
We conduct extensive experiments on both synthetic and real-world data sets,
and our results confirm the superior performance and efficiency of BKTR for
model estimation and parameter inference.
