Getis–Ord statistics explained

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

There are two different versions of the statistic, depending on whether the data point at the target location

i is included or not^[6] G _{i} = (\sum _{j} w _{ij} x _{j})/(\sum _{j} x _{j}) G ^{*} _{i} = (\sum _{j} w _{ij} x _{j})/(\sum _{j} x _{j})

Here

x _{i} is the value observed at the i ^{th} spatial site and w _{ij} is the spatial weight matrix which constrains which sites are connected to one another. For 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.

Global statistics

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

G = (\sum _{ij,} w _{ij} x _{i} x _{j})/(\sum _{ij,} x _{i} x _{j}) G ^{*} = (\sum _{ij} w _{ij} x _{i} x _{j})/(\sum _{ij} x _{i} x _{j})

The local and global

G ^{*} statistics are related through the weighted average(\sum _{i} x _{i} G ^{*} _{i})/(\sum _{i} x _{i}) = (\sum _{ij} x _{i} w _{ij} x _{j})/(\sum _{i} x _{i} \sum _{j} x _{j}) = G ^{*} The relationship of the G and G _{i} statistics is more complicated due to the dependence of the denominator of G _{i} on i .

Relation to Moran's I

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

I = (N)/(W) (\sum _{ij} w _{ij} (x _{i} - \bar{x)/()(x) _{j} - \bar{x})}{\sum _{i} (x _{i} - \bar{x}) ^{2}} where N is the number of spatial sites and W = \sum _{ij} w _{ij} is the sum of the entries in the spatial weight matrix. Getis and Ord show^[7] thatI = (K _{1/K} _{2)} G - K _{2} \bar{x} \sum _{i} (w _{i} + w _{⋅} ) x _{i} + K _{2} \bar{x} ^{2} W Where w _{i} = \sum _{j} w _{ij} , w _{⋅} = \sum _{j} w _{ji} , K _{1} = \left( \sum _{ij,} x _{i} x _{j} \right) ^{-1} and K _{2} = (W)/(N) \left(\sum _{i} (x _{i} - \bar{x}) ^{2\right)} ^{-1} . They are equal if 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} I = (1)/(W) \left( \sum _{i} z _{i} V _{i} G ^{*} _{i} - N\right) where z _{i} = (x _{i} - \bar{x})/s , s is the standard deviation of x andV ^{2} _{i} = (1)/(N -1) \sum _{j} \left( w _{ij} - (1)/(N) \sum _{k} w _{ik} \right) ^{2} is an estimate of the variance of w _{ij} .

See also

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

Notes and References

1. Web site: RPubs - R Tutorial: Hotspot Analysis Using Getis Ord Gi .
2. Web site: Hot Spot Analysis (Getis-Ord Gi*) (Spatial Statistics)—ArcGIS Pro | Documentation .
3. https://pysal.org/
4. Web site: R-spatial/Spdep . .
5. 10.1007/s11749-018-0599-x . R.S. . Bivand . D.W. . Wong . 2018 . Comparing implementations of global and local indicators of spatial association. . Test . 27 . 3 . 716–748. 11250/2565494 . free .
6. Web site: Local Spatial Autocorrelation (2) .
7. 10.1111/j.1538-4632.1992.tb00261.x . A. . Getis . J.K. . Ord . 1992 . The analysis of spatial association by use of distance statistics . . 24 . 3 . 189–206.
8. 10.1111/j.1538-4632.1995.tb00912.x . A. . Getis . J.K. . Ord . 1995 . Local spatial autocorrelation statistics: distributional issues and an application . . 27 . 4 . 286–306.
9. Web site: How High/Low Clustering (Getis-Ord General G) works—ArcGIS Pro | Documentation .