用条件极值法推导Moran指数的两组界
Deriving two sets of bounds of Moran's index by conditional extremum method
AI总结：
针对Moran指数边界值存在争议的问题，提出基于二次型条件极值的推导方法，得到两组边界值并取交集确定其数值范围，明确其界由规模向量和空间权重矩阵决定，解决了空间自相关分析的基础问题。
摘要：
Moran指数是空间自相关的基本度量指标，已被应用于自然科学与社会科学的多个领域。一个优良的度量指标应当具备明确的边界值或临界值，但目前Moran指数的边界值与临界值均存在争议。本文提出一种推导Moran指数边界值的新方法，其核心是基于Moran指数定义的二次型求解条件极值。由此自然推导出Moran指数的两组边界值：一组由空间权重矩阵的特征值决定，另一组由空间自相关系数的二次型决定（-1<Moran's I<1）。两组边界值取交集后得到Moran指数的四种可能数值范围。研究得出结论：Moran指数的界由规模向量和空间权重矩阵共同决定，其基本边界值为-1和1；空间权重矩阵的特征值代表n个地理要素的特征向量轴在不同方向上的最大延伸长度。本研究解决了空间自相关分析的基础问题之一。
英文摘要：
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.
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