  This article concerns the predictive modeling for spatio-temporal data as
well as model interpretation using data information in space and time. We
develop a novel approach based on supervised 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
emphasizes the exploration of geometric information from the data. The method
of Pairwise Directions Estimation (PDE; Lue, 2019) is implemented in our
approach as a data-driven function searching for spatial patterns and temporal
trends. The benefit of using geometric information from the method of PDE is
highlighted, which aids effectively in exploring data structures. We further
enhance PDE, referring to it as PDE+, by incorporating kriging to estimate the
random effects not explained in the mean functions. Our proposal can not only
increase prediction accuracy, but also improve the interpretation for modeling.
Two 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 and trends for
spatio-temporal data. Illustrative applications to two real datasets are also
presented.
