Urban Dynamics, Fractals and Generalized Entropy

Encarnação, Sara; Gaudiano, Marcos; Santos, Francisco C.; Tenedório, José António; Pacheco, Jorge M.

doi:10.3390/e15072679

Cited by 13 publications

(11 citation statements)

References 17 publications

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“…Berry (2005) definitely contributed to the debate perceiving cities as spatial systems, viewing the urban theory as one aspect of the General Systems Theory. Literature centered on both complex urban systems, fractal cities and entropy-based approaches grew rapidly in the last decades (see, among others, Allen, 1997;Anderson, Arrow, & Pines, 1988;Bura, Guerin-Pace, Mathian, Pumain, & Sanders, 1996;Page, Parisel, Pumain, &Sanders, 2001 for complex systems andCabral, Augusto, Tewolde, &Araya, 2013;Encarnação, Gaudiano, Santos, Tened orio, & Pacheco, 2013 for entropy-based approaches).…”

Section: Introductionmentioning

confidence: 99%

Lost in translation, found in entropy: An exploratory data analysis of latent growth factors in a Mediterranean city (1960–2010)

Salvati¹,

Mavrakis

Serra

et al. 2015

Applied Geography

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Section: Introductionmentioning

confidence: 99%

Lost in translation, found in entropy: An exploratory data analysis of latent growth factors in a Mediterranean city (1960–2010)

Salvati¹,

Mavrakis

Serra

et al. 2015

Applied Geography

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“…The measurement of spatial entropy has a natural connection with the fractal dimensions of urban systems. In the literature, both entropy and fractal dimensions have been employed to characterize urban patterns and evolution process [ 47 , 48 , 49 , 50 ]. However, the scale dependence of generalized spatial entropies and their relationships with multifractal dimension spectrums are rarely reported.…”

Section: Discussionmentioning

confidence: 99%

Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

Chen

Huang

2018

Entropy

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A type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimension can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connect between entropy and fractal dimension, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurement into fractal dimension for the spatial analysis of scale-free urban phenomena using ideas from scaling. Urban systems proved to be random prefractal and multifractals systems. The entropy of fractal cities bears two typical properties. One is the scale dependence. Entropy values of urban systems always depend on the scales of spatial measurement.The other is entropy conservation. Different fractal parts bear the same entropy value. Thus entropy cannot reflect the spatial heterogeneity of fractal cities in theory. If we convert the generalized entropy into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. The essence of scale dependence is the scaling in cities, and the spatial heterogeneity of cities can be characterized by multifractal scaling. This study may be helpful for the students to describe and understand spatial complexity of cities.

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“…Fractal geometry provides a powerful tool of scaling analysis for complex systems, and it is useful in both theoretical and empirical researches on cities. A number of studies on fractal cities have verified the effect and power of fractal methods and fractal parameters [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 ]. Unfortunately, new problems have arisen in recent years, that is, the results of fractal dimension estimation for urban form depend on scope of study area and methods of fractal dimension measurement and calculation [ 11 , 12 , 13 , 14 , 15 , 16 , 26 ].…”

Section: Introductionmentioning

confidence: 99%

The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

Chen

2019

Entropy

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Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows: First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who takes interest in or even have already participated in the studies of fractal cities.

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Urban Dynamics, Fractals and Generalized Entropy

Cited by 13 publications

References 17 publications

Lost in translation, found in entropy: An exploratory data analysis of latent growth factors in a Mediterranean city (1960–2010)

Lost in translation, found in entropy: An exploratory data analysis of latent growth factors in a Mediterranean city (1960–2010)

Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

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