Reconstruction and Normalization of Anselin's Local Indicators of Spatial Association (LISA)

TL;DR

This paper identifies and corrects a mathematical fault in Anselin's LISA measures, proposing reconstructed formulas that satisfy key theoretical requirements and validating them with empirical data from China.

Contribution

It reconstructs the calculation formulas of local Moran and Geary indicators to ensure they meet fundamental theoretical properties, clarifying misunderstandings in geospatial analysis.

Findings

The first set of LISAs satisfies Anselin's second requirement.

The new third set of LISAs are canonical forms based on normalized weights.

Empirical data from China verifies the theoretical corrections.

Abstract

The local indicators of spatial association (LISA) are significant measures for spatial autocorrelation analysis. However, there is an inadvertent fault in Anselin's mathematical processes so that the local Moran and Geary indicators do not satisfy his second basic requirement, i.e., the sum of the local indicators is proportional to a global indicator. Based on Anselin's original intention, this paper is devoted to reconstructing the calculation formulae of the local Moran indexes and Geary coefficients through mathematical derivation and empirical evidence. Two sets of LISAs were clarified by mathematical reasoning. One set of LISAs is based on no normalized weights and centralized variable (MI1 and GC1), and the other set is but the second the set cannot. Then, the third set of LISA was proposed, treated as canonical forms (MI3 and GC3). The local Moran indexes are based on global normalized weights and standardized variable based on population standard deviation, while the local Geary coefficients are based on global normalized weights and standardized variable based on sample standard deviation. This set of LISAs satisfies the second requirement of based on row normalized weights and standardized variable (MI2 and GC2). The results show that the first set of LISAs satisfy Anselin's second requirement,Anselin's. The observational data of city population and traffic mileage in Beijing-Tianjin-Hebei region of China were employed to verify the theoretical results. This study helps to clarify the misunderstandings about LISAs in the field of geospatial analysis.