Getis–Ord statistics, also known as Gi*, 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 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 and R.

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 i {\displaystyle i} is included or not

G i = ∑ j ≠ i w i j x j ∑ j ≠ i x j {\displaystyle G_{i}={\frac {\sum _{j\neq i}w_{ij}x_{j}}{\sum _{j\neq i}x_{j}}}}

G i ∗ = ∑ j w i j x j ∑ j x j {\displaystyle G_{i}^{*}={\frac {\sum _{j}w_{ij}x_{j}}{\sum _{j}x_{j}}}}

Here x i {\displaystyle x_{i}} is the value observed at the i t h {\displaystyle i^{th}} spatial site and w i j {\displaystyle w_{ij}} is the spatial weight matrix which constrains which sites are connected to one another. For G i ∗ {\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 however in practice permutation testing is used to obtain more reliable estimates of significance for statistical inference.

Global statistics

The Getis-Ord statistics of overall spatial association are

G = ∑ i j , i ≠ j w i j x i x j ∑ i j , i ≠ j x i x j {\displaystyle G={\frac {\sum _{ij,i\neq j}w_{ij}x_{i}x_{j}}{\sum _{ij,i\neq j}x_{i}x_{j}}}}

G ∗ = ∑ i j w i j x i x j ∑ i 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 G ∗ {\displaystyle G^{*}} statistics are related through the weighted average

∑ i x i G i ∗ ∑ i x i = ∑ i j x i w i j x j ∑ i x i ∑ j x j = G ∗ {\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 G {\displaystyle G} and G i {\displaystyle G_{i}} statistics is more complicated due to the dependence of the denominator of G i {\displaystyle G_{i}} on i {\displaystyle i}.

Relation to Moran's I

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

I = N W ∑ i j w i j ( x i − x ¯ ) ( x j − x ¯ ) ∑ i ( x i − x ¯ ) 2 {\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 N {\displaystyle N} is the number of spatial sites and W = ∑ i j w i j {\displaystyle W=\sum _{ij}w_{ij}} is the sum of the entries in the spatial weight matrix. Getis and Ord show that

I = ( K 1 / K 2 ) G − K 2 x ¯ ∑ i ( w i ⋅ + w ⋅ i ) x i + K 2 x ¯ 2 W {\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 w i ⋅ = ∑ j w i j {\displaystyle w_{i\cdot }=\sum _{j}w_{ij}}, w ⋅ i = ∑ j w j i {\displaystyle w_{\cdot i}=\sum _{j}w_{ji}}, K 1 = ( ∑ i j , i ≠ j x i x j ) − 1 {\displaystyle K_{1}=\left(\sum _{ij,i\neq j}x_{i}x_{j}\right)^{-1}} and K 2 = W N ( ∑ i ( x i − x ¯ ) 2 ) − 1 {\displaystyle K_{2}={\frac {W}{N}}\left(\sum _{i}(x_{i}-{\bar {x}})^{2}\right)^{-1}}. They are equal if w i j = w {\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 G i ∗ {\displaystyle G_{i}^{*}}

I = 1 W ( ∑ i z i V i G i ∗ − N ) {\displaystyle I={\frac {1}{W}}\left(\sum _{i}z_{i}V_{i}G_{i}^{*}-N\right)}

where z i = ( x i − x ¯ ) / s {\displaystyle z_{i}=(x_{i}-{\bar {x}})/s}, s {\displaystyle s} is the standard deviation of x {\displaystyle x} and

V i 2 = 1 N − 1 ∑ j ( w i j − 1 N ∑ k w i k ) 2 {\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 w i j {\displaystyle w_{ij}}.

See also