Getis–Ord statistics

Global statistics

The Getis-Ord statistics of overall spatial association are

${\displaystyle G=\frac{\sum _{ij,i\ne j}{w}_{ij}{x}_i{x}_j}{\sum _{ij,i\ne j}{x}_i{x}_j}}$

${\displaystyle G^{\ast }=\frac{\sum _{ij}{w}_{ij}{x}_i{x}_j}{\sum _{ij}{x}_i{x}_j}}$

The local and global ${\displaystyle G^{\ast }}$ statistics are related through the weighted average

${\displaystyle \frac{\sum _i{x}_i{G}_i^{\ast }}{\sum _i{x}_i}=\frac{\sum _{ij}{x}_i{w}_{ij}{x}_j}{\sum _i{x}_i\sum _j{x}_j}=G^{\ast }}$

The relationship of the ${\displaystyle G}$ and ${\displaystyle G_i}$ statistics is more complicated due to the dependence of the denominator of ${\displaystyle G_i}$ on ${\displaystyle i}$.

Relation to Moran's I

Moran's I is another commonly used measure of spatial association defined by

${\displaystyle I=\frac{N}{W}\frac{\sum _{ij}{w}_{ij}(x_i-\overline{x})(x_j-\overline{x})}{\sum _i(x_i-\overline{x}{)}^2}}$

where ${\displaystyle N}$ is the number of spatial sites and ${\displaystyle W=\sum _{ij}{w}_{ij}}$ is the sum of the entries in the spatial weight matrix. Getis and Ord show that

${\displaystyle I=(K_1/K_2)G-K_2\overline{x}\sum _i(w_{i\cdot }+w_{\cdot i})x_i+K_2{\overline{x}}^{2}W}$

Where ${\displaystyle w_{i\cdot }=\sum _j{w}_{ij}}$, ${\displaystyle w_{\cdot i}=\sum _j{w}_{ji}}$, ${\displaystyle K_1={\left(\sum _{ij,i\ne j}{x}_i{x}_j\right)}^{-1}}$ and ${\displaystyle K_2=\frac{W}{N}{\left(\sum _i(x_i-\overline{x}{)}^2\right)}^{-1}}$. They are equal if ${\displaystyle w_{ij}=w}$ is constant, but not in general.

Ord and Getis also show that Moran's I can be written in terms of ${\displaystyle G_i^{\ast }}$

${\displaystyle I=\frac{1}{W}\left(\sum _i{z}_i{V}_i{G}_i^{\ast }-N\right)}$

where ${\displaystyle z_i=(x_i-\overline{x})/s}$, ${\displaystyle s}$ is the standard deviation of ${\displaystyle x}$ and

${\displaystyle V_i^2=\frac{1}{N-1}\sum _j{\left(w_{ij}-\frac{1}{N}\sum _k{w}_{ik}\right)}^2}$

is an estimate of the variance of ${\displaystyle w_{ij}}$.