Abstract
A number of spatial statistic measurements such as Moran's I and Geary's C can be used for spatial autocorrelation analysis. Spatial autocorrelation modeling proceeded from the 1-dimension autocorrelation of time series analysis, with time lag replaced by spatial weights so that the autocorrelation functions degenerated to autocorrelation coefficients. This paper develops 2-dimensional spatial autocorrelation functions based on the Moran index using the relative staircase function as a weight function to yield a spatial weight matrix with a displacement parameter. The displacement bears analogy with the time lag in time series analysis. Based on the spatial displacement parameter, two types of spatial autocorrelation functions are constructed for 2-dimensional spatial analysis. Then the partial spatial autocorrelation functions are derived by using the Yule-Walker recursive equation. The spatial autocorrelation functions are generalized to the autocorrelation functions based on Geary's coefficient and Getis' index. As an example, the new analytical framework was applied to the spatial autocorrelation modeling of Chinese cities. A conclusion can be reached that it is an effective method to build an autocorrelation function based on the relative step function. The spatial autocorrelation functions can be employed to reveal deep geographical information and perform spatial dynamic analysis, and lay the foundation for the scaling analysis of spatial correlation.
Highlights
Measuring spatial autocorrelation is an important method for quantitative analyses in geography
This paper develops 2-dimensional spatial autocorrelation functions based on Moran’s index and the corresponding analytical process, laying the foundation for scaling analysis based on spatial autocorrelation
If the diagonal elements are taken into account, there is no significant difference between the spatial autocorrelation function (ACF) and to the spatial partial autocorrelation function (PACF)
Summary
Measuring spatial autocorrelation is an important method for quantitative analyses in geography. This method can be treated as a cornerstone of spatial statistics. In the theoretical aspect, the scaling property of geographical spatial autocorrelation has not been emphasized. Conventional mathematical modeling and quantitative analysis depend on characteristic scales. If and only if we find the valid characteristic scales such as determinate length, eigenvalue, and mean, will we be able to make effective mathematical models. If a geographical distribution is a scale-free distribution, no characteristic scale can be found, and the conventional mathematical methods will be ineffective.
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