Local and Global Spatial Statistics

Synonyms

Getis-Ord index G*; LISA statistics; Local indicators of spatial association statistics; Local Moran’s I; Statistics, spatial

Definition

These statistics assess the clustering of spatial data at the local level (using local clusters) or globally (using all the available data).

Moran’s I and Geary’s C are global indices of spatial association that include all the locations in the data. 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)

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

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