Regionalization of multiscale spatial processes by using a criterion for spatial aggregation error

Summary   The modifiable areal unit problem and the ecological fallacy are known problems that occur when modelling 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, which we minimize to obtain an optimal regionalization. To define the criterion we draw a connection between spatial aggregation error and a new multiscale representation of the Karhunen–Loève expansion. This relationship between the criterion for spatial aggregation error and the multiscale Karhunen–Loève 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 data sets: one using the American Community Survey and one related to environmental ocean winds.