Scaling invariance of spatial autocorrelation in urban built-up area

TL;DR

This paper reveals that the spatial autocorrelation in urban built-up areas follows a scale-invariant power law related to fractal dimensions, challenging traditional scale-dependent analysis methods.

Contribution

It establishes a theoretical model linking Moran's I to fractal dimensions and empirically validates the scale invariance of spatial autocorrelation in cities.

Findings

Spatial autocorrelation exhibits no characteristic scale in many urban areas.

Moran's I follows a power law with respect to measurement scale, related to fractal dimension.

Urban spatial analysis should consider scale-invariant properties rather than fixed scales.

Abstract

City is proved to be a scale-free phenomenon, and spatial autocorrelation is often employed to analyze spatial redundancy of cities. Unfortunately, spatial analysis results deviated practical requirement in many cases due to fractal nature of cities. This paper is devoted to revealing the internal relationship between the scale dependence of Moran's I and fractal scaling. Mathematical reasoning and empirical analysis are employed to derive and test the model on the scale dependence of spatial autocorrelation. The data extraction way for fractal dimension estimation is box-counting method, and parameter estimation relies on the least squares regression. In light of 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. An empirical analysis is made to testify the theoretical model. It can be concluded that spatial autocorrelation of urban built-up area has no characteristic scale in many cases, and urban spatial analysis need new thinking.