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Spatial analysis of cities using Renyi entropy and fractal parameters

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  • Chen, Yanguang
  • Feng, Jian

Abstract

The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Euclidean and fractal geometries. The main method is mathematical derivation and empirical analysis, and the theoretical foundation is the discovery that the normalized fractal dimension is equal to the normalized entropy. Based on this finding, a set of useful spatial indexes termed “generalized multifractal indicators” are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The generalized multifractal indexes are applied to the population density distribution of Hangzhou city, China. The calculation results reveal the feature of spatio-temporal evolution of Hangzhou's urban morphology. This study indicates that fractal dimension and spatial entropy can be combined to produce a new methodology for spatial analysis of city development.

Suggested Citation

  • Chen, Yanguang & Feng, Jian, 2017. "Spatial analysis of cities using Renyi entropy and fractal parameters," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 279-287.
    • Handle: RePEc:eee:chsofr:v:105:y:2017:i:c:p:279-287
    DOI: 10.1016/j.chaos.2017.10.018
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    References listed on IDEAS

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    1. repec:cai:popine:popu_p1998_10n1_0240 is not listed on IDEAS
    2. Chen, Yanguang, 2014. "Multifractals of central place systems: Models, dimension spectrums, and empirical analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 402(C), pages 266-282.
    3. Chen, Yanguang, 2012. "Fractal dimension evolution and spatial replacement dynamics of urban growth," Chaos, Solitons & Fractals, Elsevier, vol. 45(2), pages 115-124.
    4. Y. Bar-Yam, 2004. "Multiscale Complexity/Entropy," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 47-63.
    5. Chen, Yanguang & Feng, Jian, 2012. "Fractal-based exponential distribution of urban density and self-affine fractal forms of cities," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1404-1416.
    6. Chen, Yanguang & Jiang, Shiguo, 2009. "An analytical process of the spatio-temporal evolution of urban systems based on allometric and fractal ideas," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 49-64.
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    Cited by:

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    2. Saeedimoghaddam, Mahmoud & Stepinski, T.F. & Dmowska, Anna, 2020. "Rényi’s spectra of urban form for different modalities of input data," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    3. Saeedimoghaddam, Mahmoud & Stepinski, T.F., 2021. "Multiplicative random cascade models of multifractal urban structures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 569(C).

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