A geocomputation and geovisualization comparison of Moran and Geary eigenvector spatial filtering

To date, the basis of eigenvector spatial filtering is solely the centered spatial weights matrix (SWM) that is part of the Moran Coefficient (MC). The Laplacian matrix, < CI >_diag-C offers an alternative transformation of the SWM, one associated with the Geary Ratio (GR), another popular measure of spatial autocorrelation (SA). This paper compares these two specifications, respectively denoted by MCESF and GRESF. We exploit the properties that the extreme values of the MC are determined by the second and the n^th eigenvalues of the MC-based SWM, whereas the limits of the GR are a function of the largest and second smallest Laplacian eigenvalues. Both MC-based and GR-based eigenvectors share the same desirable properties of orthogonality and summing to zero (i.e., uncorrelatedness). Because the centered SWM and its Laplacian matrix counterpart partition a given SA space differently, MCESF captures more positive SA than its GRESF counterpart. The GRESF also tends to confuse positive and negative SA. The GRESF is more closely related to the row-standardized SWM W, whereas the MCESF is more closely tied to the binary SWM C. We conclude that the MCESF should furnish more parsimonious representations of SA, and hence a preferred technique for spatial analysis.