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.

This statistic was developed by Roy C. Geary.