  Dimension reduction is often the first step in statistical modeling or
prediction of multivariate spatial data. However, most existing dimension
reduction techniques do not account for the spatial correlation between
observations and do not take the downstream modeling task into consideration
when finding the lower-dimensional representation. We formalize the closeness
of approximation to the original data and the utility of lower-dimensional
scores for downstream modeling as two complementary, sometimes conflicting,
metrics for dimension reduction. We illustrate how existing methodologies fall
into this framework and propose a flexible dimension reduction algorithm that
achieves the optimal trade-off. We derive a computationally simple form for our
algorithm and illustrate its performance through simulation studies, as well as
two applications in air pollution modeling and spatial transcriptomics.
