Scaling Invariance of Spatial Autocorrelation in Urban Landscape

Abstract

Abstract Context A large number of studies have shown that measures of spatial autocorrelation in urban landscape bear scale dependence. Scale dependence generally suggests fractals and scaling. But the inherent relationships between spatial autocorrelation index and fractal dimension is not clear yet. Objectives This paper is devoted to revealing the internal relationship between the scale dependence of Moran’s I and fractal scaling. Methods Mathematical reasoning and empirical analysis are employed to derive and test the model on the scale dependence of spatial autocorrelation. The fractal dimension estimation is based on box-counting method, and parameter estimation relies on the least squares regression. Results Based on the locality postulate of spatial correlation and the idea of multifractals, a power law model on Moran’s I changing with measurement scale is derived from the principle of recursive subdivision of space. The power exponent is proved to be a function of fractal dimension. This suggests that the numerical relationship between Moran’s I and fractal dimension can be established through the scaling process of granularity. Taking the cities of Beijing, Shanghai and Guangzhou in China as examples, we made an empirical analysis with observation data, and the results lend support to the theoretical model. Conclusions It can be concluded that spatial autocorrelation of urban landscape has no characteristic scale in many cases. In order to make effective spatial statistical analysis, it is necessary to explore spatial complexity based on fractal scaling behind urban landscape.