Comparison of Moran's I and Geary's c in Multivariate Spatial Pattern Analysis

Abstract

This article compares multivariate spatial analysis methods that include not only multivariate covariance, but also spatial dependence of the data explicitly and simultaneously in model design by extending two univariate autocorrelation measures, namely Moran's I and Geary's c. The results derived from the simulation datasets indicate that the standard Moran component analysis is preferable to Geary component analysis as a tool for summarizing multivariate spatial structures. However, the generalized Geary principal component analysis developed in this study by adding variance into the optimization criterion and solved as a trace ratio optimization problem performs as well as, if not better than its counterpart the Moran principal component analysis does. With respect to the sensitivity in detecting subtle spatial structures, the choice of the appropriate tool is dependent on the correlation and variance of the spatial multivariate data. Finally, the four techniques are applied to the Social Determinants of Health dataset to analyze its multivariate spatial pattern. The two generalized methods detect more urban areas and higher autocorrelation structures than the other two standard methods, and provide more obvious contrast between urban and rural areas due to the large variance of the spatial component.