  Spatial autocorrelation plays an important role in geographical analysis,
however, there is still room for improvement of this method. The formula for
Moran's index is complicated, and several basic problems remain to be solved.
Therefore, I will reconstruct its mathematical framework using mathematical
derivation based on linear algebra and present four simple approaches to
calculating Moran's index. Moran's scatterplot will be ameliorated, and new
test methods will be proposed. The relationship between the global Moran's
index and Geary's coefficient will be discussed from two different vantage
points: spatial population and spatial sample. The sphere of applications for
both Moran's index and Geary's coefficient will be clarified and defined. One
of theoretical findings is that Moran's index is a characteristic parameter of
spatial weight matrices, so the selection of weight functions is very
significant for autocorrelation analysis of geographical systems. A case study
of 29 Chinese cities in 2000 will be employed to validate the innovatory models
and methods. This work is a methodological study, which will simplify the
process of autocorrelation analysis. The results of this study will lay the
foundation for the scaling analysis of spatial autocorrelation.
