基于Moran统计量的空间自相关理论发展和方法改进

本文旨在发展基于Moran指数的空间自相关分析理论和方法。首先,利用线性代数知识对基于Moran统计量的空间自相关过程的数学表示进行规范化整理;其次,基于变换中的不变性思想给出Moran指数的理论解释;第三,对空间权重矩阵的数理性质、建设方法和应用范围提出新的见解。总结并发展了Moran指数的三种计算方法——三步求值法、矩阵标度法和回归分析法,将空间权重矩阵划分为四种基本类型——局域关联型、准局域关联型、准长程关联型和长程关联型。以河南省鹤壁市乡镇体系为实证对象,以本文改进的理论和方法为依据,提供了一个空间自相关分析的简明案例。

Reconstructing the mathematical process of spatial autocorrelation based on Moran’s statistics

This paper is devoted to developing the theory and methods of spatial autocorrelation analysis based on the Moran statistics. Firstly, the mathematical process of the Moran’s index is reconstructed with the theory of linear algebra. Two kinds of generalized spatial weighting matrix (GSWM) are defined as follows: one is the ideal spatial weighting matrix (ISWM), and the other is the real spatial weighting matrix (RSWM). The Moran’s I can be redefined by both ISWM and RSWM. Secondly, the theoretical essence of Moran’s I is brought to light by using the ideas from symmetry and invariance of mathematical transform. The Moran’s I is in fact the eigenvalue of ISWM and RSWM, and the corresponding eigenvector is just the vector consisting of the standardized data for spatial autocorrelation analysis. Thirdly, the Moran scatterplot is revised. Based on ISWM and RSWM, the Moran scatterplot for local analysis of spatial association is improved and the result is more satisfying than the original form. In the improved scatterplot, ISWM presents a straight line, and RSWM shows itself as a random distribution of data points. Three approaches to estimating the Moran’s I are advanced as follows: (1) The method of formula. Three-step computation process is summed up by means of matrix theory. (2) The method of matrix. The scaling relation is employed to estimate the Moran’s I by calculating the eigenvalue of ISWM or RSWM. (3) The method of regression analysis. This approach is based on the correlation between the standardized vector and ISWM or RSWM. The key step of making analysis of spatial autocorrelation is to construct the contiguity matrix. The spatial weighting matrix (SWM) is divided into four types: (1) locality correlation, (2) quasi-locality correlation, (3) quasi-long-distance correlation, and (4) long distance correlation associated with action at a distance. Different types of SWM are suitable for different cases of geographical analysis. The improved theory and method of spatial autocorrelation based on the Moran’s I is applied to the systems of towns in Hebi Prefecture of Henan Province, China. Based on the measure of total population of towns, a symmetrical pattern of spatial autocorrelation, which looks like a butterfly, is revealed and illustrated. This example shows how to make use of spatial autocorrelation theory in human geographical analysis easily and simply.