Abstract
Analyses of urban scaling laws assume that observations in different cities are independent of the existence of nearby cities. Here we introduce generative models and data-analysis methods that overcome this limitation by modelling explicitly the effect of interactions between individuals at different locations. Parameters that describe the scaling law and the spatial interactions are inferred from data simultaneously, allowing for rigorous (Bayesian) model comparison and overcoming the problem of defining the boundaries of urban regions. Results in five different datasets show that including spatial interactions typically leads to better models and a change in the exponent of the scaling law.
Highlights
One of the pillars of the study of cities as complex systems is the existence of statistical laws that apply “universally” to urban regions in different locations [1,2,3,4]
Spatial interactions in urban scaling laws scrutiny is being applied to the methods used in scaling laws in urban systems [13, 14, 19,20,21] and reveal the limitations of the traditional linear-fitting approach: it relies on several simplifying assumptions, it is unable to deal with y = 0 in the data, it makes it difficult to compare to alternative models and to assess whether the scaling is non-linear (β 61⁄4 1), and it treats each city so that results are sensitive to cut-offs and fluctuations in the data of the many small cities
Spatial interactions beyond city limits can be incorporated using more general functions a (d). We start this investigation with functions a(d;α) that depend on a single parameter α that is measured in the same units of d and sets a scale for spatial interactions such that a(α;α) = 1/2
Summary
One of the pillars of the study of cities as complex systems is the existence of statistical laws that apply “universally” to urban regions in different locations [1,2,3,4]. Spatial interactions in urban scaling laws scrutiny is being applied to the methods used in scaling laws in urban systems [13, 14, 19,20,21] and reveal the limitations of the traditional linear-fitting approach: it relies on several simplifying assumptions, it is unable to deal with y = 0 in the data, it makes it difficult to compare to alternative models and to assess whether the scaling is non-linear (β 61⁄4 1), and it treats each city so that results are sensitive to cut-offs and fluctuations in the data of the many small cities. We propose the first framework to investigate scaling laws (1) that accounts simultaneously for the following three crucial points: (i) it is based on generative models (Sec. 2); (ii) It accounts for spatial interactions between different urban areas; and (iii) it allows for rigorous statistical analyses (Sec. 3), including model comparison and the inference of parameters. Results in 5 datasets from Brazil and USA show (Sec. 4) that, in most cases, models that account for spatial interactions provide a better description of the data and that the scaling exponent β depends on the spatial scaling, in agreement with previous observations [13, 14] of the dependence of β on the urban unit
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