Spatial Aggregation Entropy: A Heterogeneity and Uncertainty Metric of Spatial Aggregation

The well-known modifiable areal unit problem (MAUP) has received much attention for a long time. There still exists, however, no unified understanding and solution to the MAUP. There is not even a statistic that quantifies the effects of the MAUP. This article proposes a new metric, namely, spatial aggregation entropy (SAE), based on which the spatial heterogeneity and uncertainty of aggregated data of spatial density are defined. The SAE quantifies the changes in spatial heterogeneity and uncertainty caused by spatial aggregation. The SAE is proven to satisfy scale additivity and spatial additivity, which makes it able to quantify the MAUP effects of spatial heterogeneity and uncertainty. Furthermore, spatial additivity extends SAE to local spatial aggregation entropy (LSAE). I distinguish two types of spatial density that are associated and unassociated with area and construct their SAE mathematical formulations. In the case study, the population density and proportion of Wayne County and the state of California are explored. I calculate the SAE and LSAE of transscale spatial aggregations for the studied attributes to demonstrate their validity. The thematic maps of LSAE are made to illustrate the distribution of local spatial heterogeneity changes. Furthermore, the article compares spatial heterogeneity of the population density with its Moran’s I at different scales. Both theoretical and case studies demonstrate that the SAE could well measure spatial heterogeneity and uncertainty changes of the spatial aggregation that reflect their MAUP effects.