  This article concerns the predictive modeling for spatio-temporal data as
well as model interpretation using data information in space and time.
Intrinsically, we develop a novel approach based on dimension reduction for
such data in order to capture nonlinear mean structures without requiring a
prespecified parametric model. In addition to prediction as a common interest,
this approach focuses more on the exploration of geometric information in the
data. The method of Pairwise Directions Estimation (PDE) is incorporated in our
approach to implement the data-driven function searching of spatial structures
and temporal patterns, useful in exploring data trends. The benefit of using
geometrical information from the method of PDE is highlighted. We further
enhance PDE, referring to it as PDE+, by using resolution adaptive fixed rank
kriging to estimate the random effects not explained in the mean structures.
Our proposal can not only produce more accurate and explainable prediction, but
also increase the computation efficiency for model building. Several simulation
examples are conducted and comparisons are made with four existing methods. The
results demonstrate that the proposed PDE+ method is very useful for exploring
and interpreting the patterns of trend for spatio-temporal data. Illustrative
applications to two real datasets are also presented.
