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Equivalent relation between normalized spatial entropy and fractal dimension

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  • Chen, Yanguang

Abstract

Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy to fractal dimension is not clear in both theory and practice. This paper is devoted to revealing the new equivalence relation between spatial entropy and fractal dimension using the box-counting method. By means of a set of regular fractals, the numerical relationship between spatial entropy and fractal dimension is examined. The results show that the ratio of actual entropy (Mq) to the maximum entropy (Mmax) equals the ratio of actual dimension (Dq) to the maximum dimension (Dmax). The spatial entropy and fractal dimension of complex spatial systems can be converted into one another by using functional box-counting method. The theoretical inference is verified by observational data of urban form. A conclusion is that the normalized spatial entropy is equal to the normalized fractal dimension. Fractal dimensions proved to be the characteristic values of entropies. In empirical studies, if the linear size of spatial measurement is small enough, a normalized entropy value is infinitely approximate to the corresponding normalized fractal dimension value. Based on the theoretical result, new spatial indexes of urban space filling can be defined, and multifractal parameters can be generalized to describe both simple systems and complex systems.

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  • Chen, Yanguang, 2020. "Equivalent relation between normalized spatial entropy and fractal dimension," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    • Handle: RePEc:eee:phsmap:v:553:y:2020:i:c:s0378437120303058
    DOI: 10.1016/j.physa.2020.124627
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    1. repec:cai:popine:popu_p1998_10n1_0240 is not listed on IDEAS
    2. Chen, Yanguang, 2012. "Fractal dimension evolution and spatial replacement dynamics of urban growth," Chaos, Solitons & Fractals, Elsevier, vol. 45(2), pages 115-124.
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    4. Michael Batty & Robin Morphet & Paolo Masucci & Kiril Stanilov, 2014. "Entropy, complexity, and spatial information," Journal of Geographical Systems, Springer, vol. 16(4), pages 363-385, October.
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    2. Ortiz-Vilchis, Pilar & Lei, Mingli & Ramirez-Arellano, Aldo, 2024. "Reformulation of Deng information dimension of complex networks based on a sigmoid asymptote," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
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    4. Maaroufi, Nadir & Zerouali, El Hassan, 2023. "Point-extended box dimension," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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