Spatial autocorrelation equation based on Moran's index

Moran's index is an important spatial statistical measure used to determine the presence or absence of spatial autocorrelation, thereby determining the selection orientation of spatial statistical methods. However, Moran's index is chiefly a statistical measurement rather than a mathematical model. This paper is devoted to establishing spatial autocorrelation models by means of linear regression analysis. Using standardized vector as independent variable, and spatial weighted vector as dependent variable, we can obtain a set of normalized linear autocorrelation equations based on quadratic form and vector inner product. The inherent structure of the models' parameters are revealed by mathematical derivation. The slope of the equation gives Moran's index, while the intercept indicates the average value of standardized spatial weight variable. The square of the intercept is negatively correlated with the square of Moran's index, but omitting the intercept does not affect the estimation of the slope value. The datasets of a real urban system are taken as an example to verify the reasoning results. A conclusion can be reached that the inner product equation of spatial autocorrelation based on Moran's index is effective. The models extend the function of spatial analysis, and help to understand the boundary values of Moran's index.