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.

翻译：Moran指数是衡量空间自相关性的基本指标，已被广泛应用于自然科学和社会科学的多个领域。一个良好的指标应具有明确的边界值或临界值，然而Moran指数的边界值和临界值至今仍存在争议。本文提出了一种基于Moran指数二次型定义的条件极值求解方法，以推导其边界值。由此自然得到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）。这两组边界值的交集给出了Moran指数的四种可能数值范围。研究结论表明，Moran指数的界限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个地理要素特征向量轴在不同方向上的最大延伸长度。本工作解决了空间自相关分析中的一个基础性问题。