Hierarchical Scaling in Systems of Natural Cities

The natural city, which is essential to understand urban physical scale and identify urban sprawling in urban studies, represents the urban functional boundaries of the city defined by human activities rather than the administrative boundaries. Most studies tend to utilize physical environment data such as street networks and remote sensing data to delimitate the natural city, however, such data may not match the real distribution of human activity density in the new cities or even ghost cities in China. This paper suggests aggregating the natural city boundary from the service area polygons of points of interest based on Reilly’s Law of Retail Gravitation and the maximum entropy method, since most points of interests provide services for surrounding communities, reflecting the vitality in a bottom-up way. The results indicate that the natural city defined by points of interests shows a high resolution and accuracy, providing a method to define the natural city with POIs.

Section: Discussionmentioning

confidence: 99%

Delimitating the Natural City with Points of Interests Based on Service Area and Maximum Entropy Method

Liu

Xia

et al. 2019

“…The development of an urban system is complex system engineering, involving different spatial scales [ 46 , 47 , 48 ]. In addition to location, overall regional structure, and other related factors, the optimization of individual urban internal structure is closely related to the overall strength of regional urban system self-organization [ 49 ].…”

Section: Discussionmentioning

confidence: 99%

Spatio–Temporal Pattern of the Urban System Network in the Huaihe River Basin Based on Entropy Theory

Fan

Guo

et al. 2018

As complex systems, the spatial structure of urban systems can be analyzed by entropy theory. This paper first calculates the interaction force between cities based on the gravity model, the spatial relationship matrix between cities is constructed using the method of network modeling, and the spatial network modeling of urban system can be calculated. Secondly, the Efficiency Entropy (EE), Quality Entropy (QE), and System Entropy (SE) of urban system network are calculated and analyzed by information entropy. Finally, taking the Huaihe River Basin (HRB) as a case study, model verification and empirical analysis are performed. It is found that the spatio–temporal pattern of the urban system network structure in the basin is uneven: in space, the urban system network in the HRB presents a layer-by-layer spatial distribution centered on the core city of Xuzhou; meanwhile, the overall urban system network in the basin presents an orderly development trend. This study has certain theoretical and practical value for the planning of urban and urban systems and the coordinated development of regions.

“…Therefore, based on the variance of the error model, we propose the urban scaling law model considering heterogeneity as:

The conditional probability

in this method can be computed as:

where

Based on the definition of the maximum-likelihood fitting method, the likelihood function

can be written as:

that is:

In order to maximize the likelihood function, we log-transformed the Equation (20) as:

The joint probability density function can be expanded as:

where

denotes the prior probability of the urban population. The prior probability can be obtained by the maximum entropy statistics of the urban system [ 52 , 53 ]. However, it can be suggested that

is independent of any of estimators of urban scaling law.…”

Section: The Chvr Methodsmentioning

confidence: 99%

An Urban Scaling Estimation Method in a Heterogeneity Variance Perspective

Zhao

Tan

et al. 2019

Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this paper, a statistical regression method based on the maximum likelihood estimation considering the lower bound constraints and the heterogeneous variance of error structure, termed as CHVR, is proposed for urban scaling estimation. In the CHVR method, the heterogeneous properties of variance are explored and modeled in the form of a power-of-the-mean variance model. The maximum likelihood fitting method is supplemented to satisfy the lower bound constraints in empirical data. The CHVR method has been applied to estimating the scaling exponents of six urban attributes covering three scaling regimes in China and compared with two traditional methods. Method evaluations based on three different criteria validate that compared with both classical methods, the CHVR method is more effective and robust. Moreover, a statistical test and long-term variations of the parameter in the variance function demonstrate that the proposed heterogeneous variance function can not only describe the heterogeneity in empirical data adequately but also provide more meaningful urban information. Therefore, the CHVR method shows great potential to provide a valuable tool for effective urban scaling studies across the world and be applied to scaling law estimation in other complex systems in the future.