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

Moran's index used to be considered to come between -1 and 1. However, in recent years, some scholars argued that the boundary value of Moran's index is determined by the minimum and maximum eigenvalues of spatial weight matrix . This paper is devoted to exploring the bounds of Moran's index from a new prospective. The main analytical processes are quadratic form transformation and the method of finding conditional extremum based on quadratic form. The results show that there are at least two sets of boundary values 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 then eigenvector axes of n geographical elements at different directions.