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指数是空间自相关性的基本度量之一，被应用于自然科学和社会科学的各个领域。好的度量方式应当具备明确的临界值。但是，对于Moran指数来说，临界值是有争议的。本文提出了一种新的方法来导出Moran指数的边界值。关键在于基于定义Moran指数的二次型寻找条件极值。结果，自然推导出了Moran指数的两组边界值。其中，一个是由空间权重矩阵的特征值决定的，另一个是由空间自相关系数的二次型决定的(-1<Moran's I<1)。这两组边界值的交点给出了Moran指数的四个可能的数值范围。本文得出的结论是，Moran指数的边界值由大小向量和空间权重矩阵决定，基本边界值为-1和1。空间权重矩阵的特征值表示n个地理元素在不同方向上对应的特征向量轴的最大扩展长度。本文解决了空间自相关性分析的一个基本问题。