Deriving two sets of bounds of Moran's index by conditional extremum method

TL;DR

This paper introduces a novel method to derive the boundary values of Moran's index using conditional extremum, clarifying its critical bounds and enhancing understanding of spatial autocorrelation measures.

Contribution

A new approach based on quadratic forms and eigenvalues to determine the boundary values of Moran's index, resolving controversies about its critical bounds.

Findings

Derived two sets of boundary values for Moran's index.

Identified bounds are determined by eigenvalues and quadratic forms.

Concluded that the bounds are fundamentally -1 and 1.

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 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.