Getis–Ord statistics

File:USA Contiguous Unemployment Rate 2020.jpg

There are two different versions of the statistic, depending on whether the data point at the target location $i$ is included or not{{cite web | url=https://geodacenter.github.io/workbook/6b_local_adv/lab6b.html | title=Local Spatial Autocorrelation (2) }}

:$G\_i\; =\; \backslash frac\{\; \backslash sum\_\{j\; \backslash neq\; i\}\; w\_\{ij\}\; x\_j\}\{\backslash sum\_\{j\; \backslash neq\; i\}\; x\_j\}$

:$G\_i^*\; =\; \backslash frac\{\; \backslash sum\_\{j\}\; w\_\{ij\}\; x\_j\}\{\backslash sum\_\{j\}\; x\_j\}$

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{{Cite journal

| doi = 10.1111/j.1538-4632.1992.tb00261.x

| first1 = A.

| last1 = Getis

| first2 = J.K.

| last2 = Ord

| year = 1992

| title = The analysis of spatial association by use of distance statistics

| journal = Geographical Analysis

| volume = 24

| issue = 3

| pages = 189–206

}}{{Cite journal

| doi = 10.1111/j.1538-4632.1995.tb00912.x

| first2 = A.

| last2 = Getis

| first1 = J.K.

| last1 = Ord

| year = 1995

| title = Local spatial autocorrelation statistics: distributional issues and an application

| journal = Geographical Analysis

| volume = 27

| issue = 4

| pages = 286–306

}}

however in practice permutation testing is used to obtain more reliable estimates of significance for statistical inference.

The Getis-Ord statistics of overall spatial association are{{cite web | url=https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-statistics/h-how-high-low-clustering-getis-ord-general-g-spat.htm | title=How High/Low Clustering (Getis-Ord General G) works—ArcGIS Pro | Documentation }}

:$G\; =\; \backslash frac\{\; \backslash sum\_\{ij,\; i\backslash neq\; j\}\; w\_\{ij\}\; x\_i\; x\_j\}\{\backslash sum\_\{ij,\; i\backslash neq\; j\}\; x\_i\; x\_j\}$

:$G^*\; =\; \backslash frac\{\; \backslash sum\_\{ij\}\; w\_\{ij\}\; x\_i\; x\_j\}\{\backslash sum\_\{ij\}\; x\_i\; x\_j\}$

The local and global $G^*$ statistics are related through the weighted average

:$\backslash frac\{\; \backslash sum\_i\; x\_i\; G^*\_i\; \}\{\; \backslash sum\_i\; x\_i\; \}\; =\; \backslash frac\{\; \backslash sum\_\{ij\}\; x\_i\; w\_\{ij\}\; x\_j\; \}\{\; \backslash sum\_i\; x\_i\; \backslash sum\_j\; x\_j\; \}\; =\; G^*$

The relationship of the $G$ and $G\_i$ statistics is more complicated due to the dependence of the denominator of $G\_i$ on $i$.

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

:$$

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 $N$ is the number of spatial sites and $W\; =\; \backslash sum\_\{ij\}\; w\_\{ij\}$ is the sum of the entries in the spatial weight matrix. Getis and Ord show that

:$$

I = (K_1/K_2) G - K_2 \bar{x} \sum_i (w_{i \cdot} + w_{\cdot i}) x_i + K_2 \bar{x}^2 W

Where $w\_\{i\; \backslash cdot\}\; =\; \backslash sum\_j\; w\_\{ij\}$, $w\_\{\backslash cdot\; i\}\; =\; \backslash sum\_j\; w\_\{ji\}$, $K\_1\; =\; \backslash left(\; \backslash sum\_\{ij,\; i\backslash neq\; j\}\; x\_i\; x\_j\; \backslash right)^\{-1\}$ and $K\_2\; =\; \backslash frac\{W\}\{N\}\backslash left(\backslash sum\_\{i\}\; (x\_i\; -\; \backslash bar\{x\})^2\backslash right)^\{-1\}$. 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 = \frac{1}{W} \left( \sum_i z_i V_i G_i^* - N\right)

where $z\_i\; =\; (x\_i\; -\; \backslash bar\{x\})/s$, $s$ is the standard deviation of $x$ and

:$$

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 $w\_\{ij\}$.

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

Category:Spatial analysis

Category:Covariance and correlation

{{reflist}}