Geary's C

Geary's C is a measure of spatial autocorrelation or an attempt to determine if adjacent observations of the same phenomenon are correlated. Spatial autocorrelation is more complex than autocorrelation because the correlation is multi-dimensional and bi-directional.

Geary's C is defined as

C={\frac {(N-1)\sum _{i}\sum _{j}w_{ij}(x_{i}-x_{j})^{2}}{2W\sum _{i}(x_{i}-{\bar {x}})^{2}}}

where $N$ is the number of spatial units indexed by $i$ and $j$ ; $x$ is the variable of interest; ${\bar {x}}$ is the mean of $x$ ; $w_{ij}$ is a matrix of spatial weights with zeroes on the diagonal (i.e., $w_{ii}=0$ ); and $W$ is the sum of all $w_{ij}$ .

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. Moran's I is a measure of global spatial autocorrelation, while Geary's C is more sensitive to local spatial autocorrelation.

Geary's C is also known as Geary's contiguity ratio or simply Geary's ratio.[1]

This statistic was developed by Roy C. Geary.[2]

Sources

J. N. R. Jeffers (1973). "A Basic Subroutine for Geary's Contiguity Ratio". Journal of the Royal Statistical Society, Series D. Wiley. 22 (4). doi:10.2307/2986827.
Geary, R. C. (1954). "The Contiguity Ratio and Statistical Mapping". The Incorporated Statistician. The Incorporated Statistician. 5 (3): 115–145. doi:10.2307/2986645. JSTOR 2986645.

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[1] J. N. R. Jeffers (1973). "A Basic Subroutine for Geary's Contiguity Ratio". Journal of the Royal Statistical Society, Series D. Wiley. 22 (4). doi:10.2307/2986827.

[2] Geary, R. C. (1954). "The Contiguity Ratio and Statistical Mapping". The Incorporated Statistician. The Incorporated Statistician. 5 (3): 115–145. doi:10.2307/2986645. JSTOR 2986645.