Structural Decomposition of Moran's Index by Getis-Ord's Indices

Moran's index and Getis-Ord,s indices are important statistical measures of spatial autocorrelation analysis. Each of them has its own function and scope of application. However, the association of Moran index with Getis-Ord index is not clear. This paper is devoted to deriving and verify the relationships between Moran's index and Getis-Ord's indices using mathematical reasoning and empirical analysis. Getis-Ord's indices are employed to decompose Moran's index. The results show that there is a strict nonlinear relationship between Moran's index and Getis-Ord's indices. Moran's index consists of four components: global Getis-Ord's index, sum of local Getis-Ord's indices, number of elements, and size correlation function. Thus the mathematical structure of Moran's index is revealed. A theoretical discovery is that the characteristics of spatial autocorrelation depends on the relationship between the global Getis-Ord's index and the sum of local Getis-Ord's indices, as well as the number of spatial elements. Local Getis-Ord's indices proved to be equivalent to the potential indices based on gravity model. A conclusion can be drawn that Moran's index is related to gravity model, and thus spatial autocorrelation is associated with spatial interaction. This indicates that weak spatial interaction leads to not significant spatial autocorrelation. The conclusion is supported by observational data. This study not only helps to better understand the basic statistics of spatial analysis, but also contributes to the further development of spatial autocorrelation theory.