  The modifiable areal unit problem in geography or the change-of-support (COS)
problem in statistics demonstrates that the interpretation of spatial (or
spatio-temporal) data analysis is affected by the choice of resolutions or
geographical units used in the study. The ecological fallacy is one famous
example of this phenomenon. Here we investigate the ecological fallacy
associated with the COS problem for multivariate spatial data with the goal of
providing a data-driven discretization criterion for the domain of interest
that minimizes aggregation errors. The discretization is based on a novel
multiscale metric, called the Multivariate Criterion for Aggregation Error
(MVCAGE). Such multi-scale representations of an underlying multivariate
process are often formulated in terms of basis expansions. We show that a
particularly useful basis expansion in this context is the multivariate
Karhunen-Lo`eve expansion (MKLE). We use the MKLE to build the MVCAGE loss
function and use it within the framework of spatial clustering algorithms to
perform optimal spatial aggregation. We demonstrate the effectiveness of our
approach through simulation and through regionalization of county-level income
and hospital quality data over the United States and prediction of ocean color
in the coastal Gulf of Alaska.
