  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.
