Deriving Two Sets of Bounds of Moran’s Index by Conditional Extremum Method

Abstract

Abstract 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 parallel sets of boundary values are derived naturally for Moran’s index. One is determined by the eigenvalues of spatial weight matrix (-nλmin≤ Moran’s I ≤-nλmax), and the other is determined by the extreme values of quadratic form of spatial autocorrelation coefficient (-1≤ Moran’s I ≤1). The intersection of the 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 both size vector and spatial weight matrix. The eigenvalues of spatial weight matrix represent the maximum extension length of the eigenvector axes of geographical elements at different directions. This work solves one of the fundamental problems of spatial autocorrelation analysis.