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

$C = \frac{(N-1) \sum_{i} \sum_{j} w_{ij} (X_i-X_j)^2}{2 W \sum_{i}(X_i-\bar X)^2}$

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; and $W$ is the sum of all $w_{ij}$.

The value of Geary's C lies between 0 and 2. 1 means no spatial autocorrelation. Values lower than 1 demonstrate increasing positive spatial autocorrelation, whilst values 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

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).
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.