Geary's "C" is a measure of spatial autocorrelation that attempts to determine if observations of the same variable are spatially autocorrelated globally (rather than at the neighborhood level). Spatial autocorrelation is more complex than autocorrelation because the correlation is multi-dimensional and bi-directional.
Global Geary's C.
Geary's "C" is defined as
formula_1
where formula_2 is the number of spatial units indexed by formula_3 and formula_4; formula_5 is the variable of interest; formula_6 is the mean of formula_5; formula_8 is the formula_9 row of the spatial weights matrix formula_10 with zeroes on the diagonal (i.e., formula_11); and formula_12 is the sum of all weights in formula_10.
The value of Geary's "C" lies between 0 and some unspecified value greater than 1. Values significantly lower than 1 demonstrate increasing positive spatial autocorrelation, whilst values significantly higher than 1 illustrate increasing negative spatial autocorrelation.
Geary's "C" is inversely related to Moran's "I", but it is not identical. While Moran's "I" and Geary's "C" are both measures of global spatial autocorrelation, they are slightly different. Geary's "C" uses the sum of squared distances whereas Moran's "I" uses standardized spatial covariance. By using squared distances Geary's "C" is less sensitive to linear associations and may pickup autocorrelation where Moran's "I" may not.
Geary's "C" is also known as Geary's contiguity ratio or simply Geary's ratio.
This statistic was developed by Roy C. Geary.
Local Geary's C.
Like Moran's I, Geary's C can be decomposed into a sum of Local Indicators of Spatial Association (LISA) statistics. LISA statistics can be used to find local clusters through significance testing, though because a large number of tests must be performed (one per sampling area) this approach suffers from the multiple comparisons problem. As noted by Anselin, this means the analysis of the local Geary statistic is aimed at identifying "interesting" points which should then be subject to further investigation. This is therefore a type of exploratory data analysis.
A local version of formula_14 is given by
formula_15
where
formula_16
then,
formula_17
Local Geary's C can be calculated in GeoDa and PySAL.
Sources.
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