Spatial analysis of cities using Renyi entropy and fractal parameters

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 dummy multifractal parameters are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The dummy 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.

Recent Deforestation Pattern Changes (2000–2017) in the Central Carpathians: A Gray-Level Co-Occurrence Matrix and Fractal Analysis Approach

“…In previous studies, fractal analysis of tree cover and deforested areas were calculated by using the FFI [40,57,58], Fixed Grid 2D Lacunarity (FG2DL) [59]. Until now, fractal analysis was used in urban agglomeration, urban growth [60][61][62], or green infrastructure models of cities [63].…”

Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

“…Fractal dimension can be used to act as the characteristic parameter of urban description. This problem has been preliminarily researched in previous works [ 8 , 25 ]. In one companion paper, using the box-counting method, we revealed that spatial entropies depend on the scales of measurement, and the normalized entropy values are empirically equal to the normalized fractal dimension values [ 8 ].…”

“…Three typical fractal dimensions in global multifractal dimensions, i.e., capacity dimension D 0 , information dimension D 1 , and correlation dimension D 2 , are discussed in this research, but the results have not been generalized to multifractal parameter spectrums. In another companion paper, based on area radius scaling, the normalized Renyi entropy is generalized to multifractal spectrums [ 25 ]. Two sets of multifractal indicators are proposed to describe urban growth and form.…”

“…This suggests that the generalized correlation dimension is just the characteristic value of Renyi entropy based on spatial scales [ 25 ]. A one-parameter family of normalized measures μ ( q ) in which the probabilities in the boxes of linear size r can be constructed as follows [ 26 , 27 ]: …”

“…In fact, the commonness between entropy and fractal dimension lies in that both the entropy values and fractal dimension values depend on the method and study area. The advantage of entropy over fractal dimension is that entropy can be applied to measuring both Euclidean structures and fractal structures, while fractal dimension can only be applied to characterizing fractal structures [ 25 ]. Compared with entropies, fractal dimensions have two advantages.…”