Spatial autocorrelation plays an important role in geographical analysis,however, there is still room for improvement of this method. The formula forMoran's index is complicated, and several basic problems remain to be solved.Therefore, I will reconstruct its mathematical framework using mathematicalderivation based on linear algebra and present four simple approaches tocalculating Moran's index. Moran's scatterplot will be ameliorated, and newtest methods will be proposed. The relationship between the global Moran'sindex and Geary's coefficient will be discussed from two different vantagepoints: spatial population and spatial sample. The sphere of applications forboth Moran's index and Geary's coefficient will be clarified and defined. Oneof theoretical findings is that Moran's index is a characteristic parameter ofspatial weight matrices, so the selection of weight functions is verysignificant for autocorrelation analysis of geographical systems. A case studyof 29 Chinese cities in 2000 will be employed to validate the innovatory modelsand methods. This work is a methodological study, which will simplify theprocess of autocorrelation analysis. The results of this study will lay thefoundation for the scaling analysis of spatial autocorrelation.
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