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I'm just wondering if someone could clarify something for me regarding spatial auto-correlation. I've been reading up on Moran's I (global and local) and Gi* statistics, and saw that at the end of Anselin (1995) he writes

Firstly, the LISA generalize the idea underlying the Getis and Ord Gi and Gi* statistics to a broad class of measures of local spatial association. Secondly, by directly linking the local indicators to a global measure of spatial association, the decomposition of the latter into its observation-specific components becomes straightforward, thus enabling the assessment of influential observations and outliers. It is this dual property that distinguishes the class of LISA from existing techniques, such as the Gi and Gi* statistics and the Moran scatterplot. The LISA presented here are easy to implement and lend themselves readily to visualization. They thus serve a useful purpose in an exploratory analysis of spatial data, potentially indicating local spatial clusters and forming the basis for a sensitivity analysis (outliers). While the former is more appropriate when no global spatial autocorrelation is present, the latter is particularly useful when there is spatial autocorrelation in the data...Furthermore, the uncritical use of the null distribution in the presence of global spatial autocorrelation will give incorrect significance levels. The problem also pertains to the Gi and Gi* statistics and would suggest that a test for global spatial autocorrelation should precede the assessment of significant local spatial clusters.

I've interpreted this as essentially saying it is not appropriate to look for local indicators of spatial auto-correlation (using for e.g. Local Moran's I) in the presence of global spatial-autocorrelation, but that's it's appropriate to examine the same datasets for hot (or cold) spots using Gi*? How should you interpret significant Local Moran's I and significant Gi* statistics (regardless of whether or not P-values have been corrected) in the presence of global auto-correlation?

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