Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

Abstract

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.

Fractal Plasmons on Cantor Set Thin Film

“…The deviation from the straight line is caused by finite accuracy of box-counting dimension calculation, specifically the choice of points to fit the asymptote shown on Figure 5. This is caused by the fact that the entropy value of a fractal system depends on the scale of measurement (in our case, number of points N), but the fractal dimension is independent of the scales [16]. Finally, from Figure 9c we conclude that it is possible to estimate the structure entropy not only from its fractal dimension but also from the dimension of its reflection spectrum.…”

“…There is an inherent connection between entropy and fractal dimension [15]; recently, Chen [16] has used this connection for the study of urban systems, which are natural, self-organizing, fractal structure. While there are some studies of entropy of electromagnetic waves reflected from metallic structure [17], it seems that this paper is the first attempt to link the relevance of entropy of fractal systems where surface plasmons are excited.…”

Differences in Gating Dynamics of BK Channels in Cellular and Mitochondrial Membranes from Human Glioblastoma Cells Unraveled by Short- and Long-Range Correlations Analysis

“…The obtained results show that the spectral width is evidently higher in the case of the BK channels from mitochondrial patches than the ones from the cellular membrane, which suggests a higher complexity and entropy of the activity of the mitoBK channel [ 68 , 69 ]. All the presented differences between the mf-spectrum width of cellular and mitochondrial series reached the statistical significance for the Mann–Whitney U test ( p < 0.05).…”

The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

“…As a special example, Vicsek fractal (box growing fractal) has fractal area (aspect 1, d T = 0) and fractal boundary (aspect 2, d T = 1), both the two fractal dimension values are D = ln(5)/ln(3) = 1.465 [ 33 ]; Sierpinski gasket also has two aspects, the fractal dimension of fractal area (aspect 1, d T = 0) is D = ln(3)/ln(2) = 1.585 [ 5 ], while the fractal dimension of fractal boundary (aspect 2, d T = 1) is D = ln(5)/ln(2) = 2.3219 [ 34 ]. Please note that the similarity dimension and radial dimension can exceed the dimension value of its embedding space; in contrast, the box dimension must come between the topological dimension d T and the Euclidean dimension of the embedding space d E [ 12 , 14 , 16 , 28 , 32 ]. The Koch lake is a special case, which also has two aspects: area and perimeter.…”

“…A viewpoint is that monofractals are mostly concerned with spaces, while multifractals deal with measures [ 27 ]. In fact, both monofractals and multifractals can be unified into the same mathematical framework by the ideas from entropy conservation [ 28 ]. The remaining parts of this paper are organized as follows.…”