Spatial Autocorrelation Measures

Definition

A statistic that assesses the global clustering of spatial data.

Main Text

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 non-zero elements indicating contiguity of locations i and j are used to code proximities. The most commonly used of global indicators of spatial autocorrelation are Moran's I and Geary's C which are defined as:

$$ I = \frac{N \sum\nolimits_i \sum\nolimits_j {W_{ij} Z_i Z_j}}{\sum\nolimits_i \sum\nolimits_i{W_{ij}} \sum\nolimits_i {Z_i^2}}\:, $$

(1)

$$ C = \frac{(N-1) \sum \nolimits_i \sum \nolimits_j {W_{ij} (x_i -x_j)^2}}{2\left(\sum\nolimits_i \sum\nolimits_j{W_{ij}}\right) \sum\nolimits_i{Z_i^2}}\:. $$

(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.