  Spatial autocorrelation coefficients such as Moran's index proved to be an
eigenvalue of the spatial correlation matrixes. An eigenvalue represents a kind
of characteristic length for quantitative analysis. However, if a spatial
correlation is based on self-organized evolution, complex structure, and the
distributions without characteristic scale, the eigenvalue will be ineffective.
In this case, the single Moran index cannot lead to reliable statistic
inferences. This paper is devoted to finding advisable approach to measure
spatial autocorrelation for the scale-free processes of complex systems by
means of mathematical reasoning and empirical analysis. Based on relative step
function as spatial contiguity function, a series of ordered spatial
autocorrelation coefficients are converted into the corresponding spatial
autocorrelation functions. Then the mathematical relation between spatial
correlation dimension and spatial autocorrelation functions is derived by
decomposition of spatial autocorrelation functions. As results, a set of useful
mathematical models are constructed for spatial analysis. Using these models,
we can utilize spatial correlation dimension to make simple spatial
autocorrelation analysis, and use spatial autocorrelation functions to make
complex spatial autocorrelation analysis for geographical phenomena. This study
reveals the inherent association of fractal patterns with spatial
autocorrelation processes in nature and society. The work may inspire new ideas
of spatial modeling and analysis for complex geographical systems.
