Chen Yanguang
Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing, 100871, People's Republic of China.
Sci Rep. 2023 Nov 7;13(1):19296. doi: 10.1038/s41598-023-45947-x.
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.
莫兰指数是一种重要的空间统计量度，用于确定空间自相关的存在与否，从而确定空间统计方法的选择方向。然而，莫兰指数主要是一种统计量度，而非数学模型。本文致力于通过线性回归分析建立空间自相关模型。以标准化向量作为自变量，空间加权向量作为因变量，基于二次型和向量内积可得到一组归一化线性自相关方程。通过数学推导揭示了模型参数的内在结构。方程的斜率给出莫兰指数，而截距表示标准化空间权重变量的平均值。截距的平方与莫兰指数的平方呈负相关，但省略截距不影响斜率值的估计。以一个真实城市系统的数据集为例验证推理结果。可以得出结论，基于莫兰指数的空间自相关内积方程是有效的。这些模型扩展了空间分析的功能，并有助于理解莫兰指数的边界值。
