A Note on the Extremities of Local Moran's Iis and Their Impact on Global Moran's I

Abstract

By defining local Moran's Ii as a ratio of quadratic forms and making use of its overall additivity to match global Moran's I, we can identify spatial objects with a strong impact on global Moran's I. First, we concentrate on the spatial properties of local Moran's Ii expressed by the local linkage degree. Depending on whether we use the W‐ or C‐coding of the spatial connectivity matrix, the variance of local Moran's Ii for a small local linkage degree will be either large or small. Note that spatial objects associated with a local Moran's Ii with a large variance affect the global statistic much more than spatial objects associated with a local Moran's Ii with a small variance. Counterintuitively, global Moran's I defined in the W‐coding is most influenced by spatial objects with a small number of spatial neighbors. In contrast, spatial objects with a large number of spatial neighbors exert more impact on global Moran's I setup in the C‐coding. Second, we investigate the impact of the empirical data on local Moran's Ii and show that local Moran's Ii will only be significant for extreme absolute residuals at and around the reference location. Clusters of average regression residuals cannot be detected by local Moran's Ii. Consequently, spatial cliques of extreme residuals contribute more to significance tests on global autocorrelation.