  Probabilistic modeling of multidimensional spatiotemporal data is critical to
many real-world applications. As real-world spatiotemporal data often exhibits
complex dependencies that are nonstationary and nonseparable, developing
effective and computationally efficient statistical models to accommodate
nonstationary/nonseparable processes containing both long-range and short-scale
variations becomes a challenging task, in particular for large-scale datasets
with various corruption/missing structures. In this paper, we propose a new
statistical framework -- Bayesian Complementary Kernelized Learning (BCKL) --
to achieve scalable probabilistic modeling for multidimensional spatiotemporal
data. To effectively characterize complex dependencies, BCKL integrates two
complementary approaches -- kernelized low-rank tensor factorization and
short-range spatiotemporal Gaussian Processes. Specifically, we use a
multi-linear low-rank factorization component to capture the global/long-range
correlations in the data and introduce an additive short-scale GP based on
compactly supported kernel functions to characterize the remaining local
variabilities. We develop an efficient Markov chain Monte Carlo (MCMC)
algorithm for model inference and evaluate the proposed BCKL framework on both
synthetic and real-world spatiotemporal datasets. Our experiment results show
that BCKL offers superior performance in providing accurate posterior mean and
high-quality uncertainty estimates, confirming the importance of both global
and local components in modeling spatiotemporal data.
