Geary's C

Geary's C is a measure of spatial autocorrelation. Like autocorrelation, spatial autocorrelation means that adjacent observations of the same phenomenon are correlated. However, autocorrelation is about proximity in time. Spatial autocorrelation is about proximity in (two-dimensional) space. Spatial autocorrelation is more complex than autocorrelation because the correlation is two-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; 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. Smaller (larger) than 1 means positive (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, Geary's Ratio, or the Geary Index.

This statistic was developed by Roy C. Geary.