A Criterion for Aggregation Error for Multivariate Spatial Data

Abstract: The criterion for aggregation error (CAGE) is an important metric that aims to measure errors that arise in multiscale (or multi-resolution) spatial data, referred to as the modifiable areal unit problem and the ecological fallacy. Specifically, CAGE is a measure of between scale variance of eigenvectors in a Karhunen-Lo\'{e}ve expansion (KLE), motivated by a theoretical result, referred to as the ``null-MAUP-theorem,'' that states that the MAUP/ecological fallacy are not present when this variance is zero. CAGE was originally developed for univariate spatial data, but its use has been applied to multivariate spatial data without the development of a null-MAUP-theorem in the multivariate spatial setting. To fill this gap, we provide theoretical justification for a multivariate CAGE (MVCAGE), which includes multiscale multivariate extensions of the KLE, Mercer's theorem, and the-null-MAUP theorem. Additionally, we provide technical results that demonstrate that the MVCAGE is preferable to spatial-only CAGE, and extend commonly used basis functions used to compute CAGE to the multivariate spatial setting. Empirical results are provided to demonstrate the use of MVCAGE for uncertainty quantification and regionalization.