Spatial analysis of cities using Renyi entropy and fractal parameters

Highlights: • Multifractal measures can be generalized to describe urban density distribution. • A set of spatial indexes are constructed based on entropy and multifractal measures. • The generalized multifractal parameters are termed apparent multifractal indexes. • An empirical analysis of cities is made with the apparent multifractal 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 "generalized multifractal indicators" are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The generalized 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.