Getis–Ord statistics

Getis–Ord statistics, also known as G_i^*, are used in spatial analysis to measure the local and global spatial autocorrelation. Developed by statisticians Arthur Getis and J. Keith Ord they are commonly used for Hot Spot Analysis^[1][2] to identify where features with high or low values are spatially clustered in a statistically significant way. Getis-Ord statistics are available in a number of software libraries such as CrimeStat, GeoDa, ArcGIS, PySAL^[3] and R.^[4][5]

Local statistics

Hot spot map showing hot and cold spots in the 2020 USA Contiguous Unemployment Rate, calculated using Getis Ord Gi*

There are two different versions of the statistic, depending on whether the data point at the target location ${\displaystyle i}$ is included or not^[6]

${\displaystyle G_{i}={\frac {\sum _{j\neq i}w_{ij}x_{j}}{\sum _{j\neq i}x_{j}}}}$

${\displaystyle G_{i}^{*}={\frac {\sum _{j}w_{ij}x_{j}}{\sum _{j}x_{j}}}}$

Here ${\displaystyle x_{i}}$ is the value observed at the ${\displaystyle i^{th}}$ spatial site and ${\displaystyle w_{ij}}$ is the spatial weight matrix which constrains which sites are connected to one another. For ${\displaystyle G_{i}^{*}}$ the denominator is constant across all observations.

A value larger (or smaller) than the mean suggests a hot (or cold) spot corresponding to a high-high (or low-low) cluster. Statistical significance can be estimated using analytical approximations as in the original work^[7][8] however in practice permutation testing is used to obtain more reliable estimates of significance for statistical inference.^[6]

Global statistics

The Getis-Ord statistics of overall spatial association are^[7][9]

${\displaystyle G={\frac {\sum _{ij,i\neq j}w_{ij}x_{i}x_{j}}{\sum _{ij,i\neq j}x_{i}x_{j}}}}$

${\displaystyle G^{*}={\frac {\sum _{ij}w_{ij}x_{i}x_{j}}{\sum _{ij}x_{i}x_{j}}}}$

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

${\displaystyle {\frac {\sum _{i}x_{i}G_{i}^{*}}{\sum _{i}x_{i}}}={\frac {\sum _{ij}x_{i}w_{ij}x_{j}}{\sum _{i}x_{i}\sum _{j}x_{j}}}=G^{*}}$

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

${\displaystyle I={\frac {N}{W}}{\frac {\sum _{ij}w_{ij}(x_{i}-{\bar {x}})(x_{j}-{\bar {x}})}{\sum _{i}(x_{i}-{\bar {x}})^{2}}}}$

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^[7] that

${\displaystyle I=(K_{1}/K_{2})G-K_{2}{\bar {x}}\sum _{i}(w_{i\cdot }+w_{\cdot i})x_{i}+K_{2}{\bar {x}}^{2}W}$

Where ${\displaystyle w_{i\cdot }=\sum _{j}w_{ij}}$, ${\displaystyle w_{\cdot i}=\sum _{j}w_{ji}}$, ${\displaystyle K_{1}=\left(\sum _{ij,i\neq j}x_{i}x_{j}\right)^{-1}}$ and ${\displaystyle K_{2}={\frac {W}{N}}\left(\sum _{i}(x_{i}-{\bar {x}})^{2}\right)^{-1}}$. They are equal if ${\displaystyle w_{ij}=w}$ is constant, but not in general.

Ord and Getis^[8] also show that Moran's I can be written in terms of ${\displaystyle G_{i}^{*}}$

${\displaystyle I={\frac {1}{W}}\left(\sum _{i}z_{i}V_{i}G_{i}^{*}-N\right)}$

where ${\displaystyle z_{i}=(x_{i}-{\bar {x}})/s}$, ${\displaystyle s}$ 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 ${\displaystyle w_{ij}}$.

See also

• Spatial analysis
• Indicators of spatial association
• Tobler's first law of geography
• Moran's I
• Geary's C