论文标题
通过有条件的极值方法得出Moran指数的两组界限
Deriving two sets of bounds of Moran's index by conditional extremum method
论文作者
论文摘要
莫兰（Moran）的指数是空间自相关的基本度量，它已应用于自然科学和社会科学的各个领域。一个好的度量应具有清晰的边界值或关键值。但是，对于Moran的索引，边界值和临界价值都是有争议的。在本文中，提出了一种新方法来得出莫兰指数的边界值。关键在于基于定义莫兰指数的二次形式找到条件极值。结果，对于Moran的索引，自然得出了两组边界值。一个由空间重量矩阵的特征值确定，另一个由空间自相关系数的二次形式（-1 <moran i <1）确定。这两组边界值的相交给出了Moran索引的四个可能的数值范围。可以得出一个结论，莫兰指数的边界由大小矢量和空间重量矩阵确定，基本边界值为-1和1。空间重量矩阵的特征值代表不同方向上N地理元素的N地理元素特征向量轴的最大延伸长度。这项工作解决了空间自相关分析的基本问题之一。
Moran's index is a basic measure of spatial autocorrelation, which has been applied to varied fields of both natural and social sciences. A good measure should have clear boundary values or critical value. However, for Moran's index, both boundary values and critical value are controversial. In this paper, a novel method is proposed to derive the boundary values of Moran's index. The key lies in finding conditional extremum based on quadratic form of defining Moran's index. As a result, two sets of boundary values are derived naturally for Moran's index. One is determined by the eigenvalues of spatial weight matrix, and the other is determined by the quadratic form of spatial autocorrelation coefficient (-1<Moran's I<1). The intersection of these two sets of boundary values gives four possible numerical ranges of Moran's index. A conclusion can be reached that the bounds of Moran's index is determined by size vector and spatial weight matrix, and the basic boundary values are -1 and 1. The eigenvalues of spatial weight matrix represent the maximum extension length of the eigenvector axes of n geographical elements at different directions. This work solves one of the fundamental problems of spatial autocorrelation analysis.
