Getis–Ord statistics

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[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

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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   is included or not[6]

 
 

Here   is the value observed at the   spatial site and   is the spatial weight matrix which constrains which sites are connected to one another. For   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

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

 
 

The local and global   statistics are related through the weighted average

 

The relationship of the   and   statistics is more complicated due to the dependence of the denominator of   on  .

Relation to Moran's I

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

 

where   is the number of spatial sites and   is the sum of the entries in the spatial weight matrix. Getis and Ord show[7] that

 

Where  ,  ,   and  . They are equal if   is constant, but not in general.

Ord and Getis[8] also show that Moran's I can be written in terms of  

 

where  ,   is the standard deviation of   and

 

is an estimate of the variance of  .

See also

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References

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