Multidimensional spatial autocorrelation analysis and it’s application based on improved Moran’s I

This paper aims to improve and extend the improved spatial Moran’s I theory by analyzing multi-observation samples. By constructing an expanded spatial weight matrix, a vector definition of the improved spatial Moran’s I is given. In order to improve the judgment basis of the improved spatial Moran’s I, the range of the improved spatial Moran’s I is derived using the non-negativity of variance. Since the improved Moran’s I is only applicable to the analysis of a single variable with unknown distribution, a Moran’s I matrix suitable for analyzing the spatial autocorrelation of multiple variables is proposed. The distribution of the elements of the Moran’s I matrix is studied by Monte Carlo simulation. The simulation results show that only the elements on the non-main diagonal follow a normal distribution when the sample size is small. Any element follows a normal distribution when the sample size is large. Then it is proved that the Moran’s I matrix follows a Wishart distribution when the spatial weight matrix is a positive definite matrix. Finally, several comprehensive evaluation indicators suitable for the theory of multivariate spatial autocorrelation are proposed based on the algebraic meaning of the Moran’s I matrix. Spatial autocorrelation analysis is carried out in combination with multi-dimensional air pollution data.