Decoding the city: multiscale spatial information of urban income

Cities are characterized by the coexistence of general aggregate patterns, along with many local variations. This poses challenges for analyses of urban phe- nomena, which tend to be either too aggregated or too local, depending on the disciplinary approach. Here, we use methods from statistical learning theory to develop a general methodology for quantifying how much information is encoded in the spatial structure of cities at different scales. We illustrate the approach via the multiscale analysis of income distributions in over 900 US metropolitan areas. By treating the formation of diverse neighborhood structures as a process of spatial selection, we quantify the complexity of explanation needed to account for personal income heterogeneity observed across all US urban areas and each of their neighborhoods. We find that spatial selection is strongly dependent on income levels with richer and poorer households appearing spatially more seg- regated than middle-income groups. We also find that different neighborhoods present different degrees of income specificity and inequality, motivating analysis and theory beyond averages. Our findings emphasize the importance of multi- scalar statistical methods that both coarse-grain and fine-grain data to bridge local to global theories of cities and other complex systems.