Spatial Autocorrelation Measures

Definition

A statistic that assesses the global clustering of spatial data.

Moran’s I and Geary’s C are indices of spatial autocorrelation. A spatial contiguity matrix W _ij , with a zero diagonal, and the off-diagonal nonzero elements indicating contiguity of locations i and j are used to code proximities. The most commonly used global indicators of spatial autocorrelation are Moran’s I and Geary’s C which are defined as

$$\displaystyle\begin{array}{rcl} I& = \frac{N\sum _{i}\sum _{j}W_{ij}Z_{i}Z_{j}} {\sum _{i}\sum _{i}W_{ij}\sum _{i}Z_{i}^{2}}, &{}\end{array}$$

(1)

$$\displaystyle\begin{array}{rcl} C& = \frac{(N-1)\sum _{i}\sum _{j}W_{ij}(x_{i}-x_{j})^{2}} {2\left (\sum _{i}\sum _{j}W_{ij}\right )\sum _{i}Z_{i}^{2}}. &{}\end{array}$$

(2)

Z _i is the deviation of the variable of interest x _i from the mean $\bar{x}$ at location i, and N is the number of data points.