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Spatial autocorrelation coefficients such as Moran's index proved to be aneigenvalue of the spatial correlation matrixes. An eigenvalue represents a kindof characteristic length for quantitative analysis. However, if a spatialcorrelation is based on self-organized evolution, complex structure, and thedistributions without characteristic scale, the eigenvalue will be ineffective.In this case, the single Moran index cannot lead to reliable statisticinferences. This paper is devoted to finding advisable approach to measurespatial autocorrelation for the scale-free processes of complex systems bymeans of mathematical reasoning and empirical analysis. Based on relative stepfunction as spatial contiguity function, a series of ordered spatialautocorrelation coefficients are converted into the corresponding spatialautocorrelation functions. Then the mathematical relation between spatialcorrelation dimension and spatial autocorrelation functions is derived bydecomposition of spatial autocorrelation functions. As results, a set of usefulmathematical models are constructed for spatial analysis. Using these models,we can utilize spatial correlation dimension to make simple spatialautocorrelation analysis, and use spatial autocorrelation functions to makecomplex spatial autocorrelation analysis for geographical phenomena. This studyreveals the inherent association of fractal patterns with spatialautocorrelation processes in nature and society. The work may inspire new ideasof spatial modeling and analysis for complex geographical systems.

Derivation of Correlation Dimension from Spatial Autocorrelation Functions