Getis–Ord Statistics
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Getis–Ord statistics, also known as Gi*, are used in
spatial analysis Spatial analysis is any of the formal Scientific technique, techniques which study entities using their topological, geometric, or geographic properties, primarily used in Urban design, Urban Design. Spatial analysis includes a variety of techni ...
to measure the local and global spatial
autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
. Developed by statisticians
Arthur Getis Arthur Getis (July 6, 1934 – May 13, 2022) was an American geographer known for his significant contributions to spatial statistics and geographic information science (GIScience). With a career spanning over four decades, Getis authored more tha ...
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 CrimeStat is a crime mapping software program. CrimeStat is Windows-based program that conducts spatial and statistical analysis and is designed to interface with a geographic information system (GIS). The program is developed by Ned Levine & Assoc ...
, GeoDa,
ArcGIS ArcGIS is a family of client, server and online geographic information system (GIS) software developed and maintained by Esri. ArcGIS was first released in 1982 as ARC/INFO, a command line-based GIS. ARC/INFO was later merged into ArcGIS De ...
, PySAL and R.


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 : G_i = \frac : G_i^* = \frac Here x_i is the value observed at the i^ spatial site and w_ 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 In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
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 Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of ...
.


Global statistics

The Getis-Ord statistics of overall spatial association are : G = \frac : G^* = \frac The local and global G^* statistics are related through the weighted average : \frac = \frac = 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 In statistics, Moran's ''I'' is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran. Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space. Spatial autocorrelation is more ...
is another commonly used measure of spatial association defined by : I = \frac \frac where N is the number of spatial sites and W = \sum_ w_ is the sum of the entries in the spatial weight matrix. Getis and Ord show that : I = (K_1/K_2) G - K_2 \bar \sum_i (w_ + w_) x_i + K_2 \bar^2 W Where w_ = \sum_j w_, w_ = \sum_j w_, K_1 = \left( \sum_ x_i x_j \right)^ and K_2 = \frac\left(\sum_ (x_i - \bar)^2\right)^. They are equal if w_=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 = \frac \left( \sum_i z_i V_i G_i^* - N\right) where z_i = (x_i - \bar)/s, s is the
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
of x and : V_i^2 = \frac\sum_j \left( w_ - \frac \sum_k w_\right)^2 is an estimate of the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of w_.


See also

*
Spatial analysis Spatial analysis is any of the formal Scientific technique, techniques which study entities using their topological, geometric, or geographic properties, primarily used in Urban design, Urban Design. Spatial analysis includes a variety of techni ...
*
Indicators of spatial association Indicators of spatial association are statistics that evaluate the existence of clusters in the spatial arrangement of a given variable. For instance, if we are studying cancer rates among census tracts in a given city local clusters in the rates m ...
* Tobler's first law of geography *
Moran's I In statistics, Moran's ''I'' is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran. Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space. Spatial autocorrelation is more ...
* Geary's C Spatial analysis Covariance and correlation


References

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