  The modifiable areal unit problem and the ecological fallacy are known
problems that occur when modeling multiscale spatial processes. We investigate
how these forms of spatial aggregation error can guide a regionalization over a
spatial domain of interest. By "regionalization" we mean a specification of
geographies that define the spatial support for areal data. This topic has been
studied vigorously by geographers, but has been given less attention by spatial
statisticians. Thus, we propose a criterion for spatial aggregation error
(CAGE), which we minimize to obtain an optimal regionalization. To define CAGE
we draw a connection between spatial aggregation error and a new multiscale
representation of the Karhunen-Loeve (K-L) expansion. This relationship between
CAGE and the multiscale K-L expansion leads to illuminating theoretical
developments including: connections between spatial aggregation error, squared
prediction error, spatial variance, and a novel extension of Obled-Creutin
eigenfunctions. The effectiveness of our approach is demonstrated through an
analysis of two datasets, one using the American Community Survey and one
related to environmental ocean winds.
