Abstract
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.
Highlights
The theory of spatial autocorrelation has been a key element of geographical analysis for more than twenty years
Based on the power function, the error sum of square is around Sf = 1.2570, the standard error is about sf = 0.2082; Based on the exponential function, we have Sf = 0.1878 as well as sf = 0.0805. These results show that, for the city sizes of China in 2000, the spatial autocorrelation based on the exponential function is more significant than the correlation based on the power function
The significance of this work is that it provides a new approach to and a new way of understanding spatial autocorrelation analysis
Summary
The theory of spatial autocorrelation has been a key element of geographical analysis for more than twenty years. One is Moran’s index [15], and the other, is Geary’s coefficient [16]. The former is a generalization of Pearson’s correlation coefficient, and the latter is analogous to the Durbin-Watson statistic of regression analysis. Moran’s index is somewhat equivalent to Geary’s coefficient and they can be substituted for one another. In practice, Moran’s index cannot be replaced by Geary’s coefficient and vice versa due to a subtle difference of statistical treatment. Compared with Geary’s coefficient, Moran’s index is more significant to spatial analysis
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