# Derivation of an Inverse Spatial Autoregressive Model for Estimating Moran's Index

Spatial autocorrelation measures such as Moran's index can be expressed as a pair of equations based on a standardized size variable and a globally normalized weight matrix. One is based on inner product, and the other is based on outer product of the size variable. The inner product equation is actually a spatial autocorrelation model. However, the theoretical basis of the inner product equation for Moran's index is not clear. This paper is devoted to revealing the antecedents and consequences of the inner product equation of Moran's index. The method is mathematical derivation and empirical analysis. The main results are as follows. First, the inner product equation is derived from a simple spatial autoregressive model, and thus the relation between Moran's index and spatial autoregressive coefficient is clarified. Second, the least squares regression is proved to be one of effective approaches for estimating Moran's index and spatial autoregressive coefficient. Third, the value ranges of the Moran's index and spatial autoregressive coefficient can be identified from three angles of view. A conclusion can be drawn that spatial autocorrelation model is actually an inverse spatial autoregressive model, and Moran's index, spatial autoregressive models, and canonical Moran's scatterplot can be integrated into the same framework through inner product and outer product equations. This work may be helpful for understanding the connections and differences between spatial autocorrelation measurements and spatial autoregressive modeling.

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