  A large number of scientific studies and engineering problems involve
high-dimensional spatiotemporal data with complicated relationships. In this
paper, we focus on a type of space-time interaction named \emph{temporal
evolution of spatial dependence (TESD)}, which is a zero time-lag
spatiotemporal covariance. For this purpose, we propose a novel Bayesian
nonparametric method based on non-stationary spatiotemporal Gaussian process
(STGP). The classic STGP has a covariance kernel separable in space and time,
failed to characterize TESD. More recent works on non-separable STGP treat
location and time together as a joint variable, which is unnecessarily
inefficient. We generalize STGP (gSTGP) to introduce the time-dependence to the
spatial kernel by varying its eigenvalues over time in the Mercer's
representation. The resulting non-stationary non-separable covariance model
bares a quasi Kronecker sum structure. Finally, a hierarchical Bayesian model
for the joint covariance is proposed to allow for full flexibility in learning
TESD. A simulation study and a longitudinal neuroimaging analysis on
Alzheimer's patients demonstrate that the proposed methodology is
(statistically) effective and (computationally) efficient in characterizing
TESD. Theoretic properties of gSTGP including posterior contraction (for
covariance) are also studied.
