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.

23 pages, 3 figures, 5 tables

References in corpus (5)

• Multifractal to monofractal evolution of the London's street network
• Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension
• Defining urban and rural regions by multifractal spectrums of urbanization
• Fractal-Based Exponential Distribution of Urban Density and Self-Affine Fractal Forms of Cities
• Normalizing and classifying shape indexes of cities by ideas from fractals

Cited by in corpus (6)

• Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension
• Spatial Measures of Urban Systems: from Entropy to Fractal Dimension
• Renyi's spectra of urban form for different modalities of input data
• Complexity in patterns of racial segregation
• Modeling Urban Growth and Form with Spatial Entropy
• Derivation of Correlation Dimension from Spatial Autocorrelation Functions