# Indicators of spatial association

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'''Indicators of spatial association''' are [statistics](/source/statistics) that evaluate the existence of [clusters](/source/Cluster_analysis) in the [spatial](/source/space) arrangement of a given variable. For instance, if we are studying [cancer](/source/cancer) rates among [census tract](/source/census_tract)s in a given city local clusters in the rates mean that there are areas that have higher or lower rates than is to be expected by chance alone; that is, the values occurring are above or below those of a random distribution in space.

==Global indicators==
thumb|Spatial Autocorrelation Report generated by ArcGIS Pro for 2022 U.S. county population.
Notable global indicators of spatial association include:<ref name="Grekousis">{{Cite book |author=George Grekousis |year=2020 |title=Spatial Analysis Methods and Practice |publisher=Cambridge University Press |page=210 |isbn=9781108712934}}</ref>

* '''Global [Moran's ''I''](/source/Moran's_I)''': The most commonly used measure of global [spatial autocorrelation](/source/spatial_autocorrelation) or the overall clustering of the spatial data developed by [Patrick Alfred Pierce Moran](/source/Patrick_Alfred_Pierce_Moran).<ref>{{Cite journal | last1 = Moran | first1 = P. A. P. | author-link = Pat Moran (statistician)| title = Notes on Continuous Stochastic Phenomena | journal = Biometrika | volume = 37 | issue = 1 | pages = 17–23 | doi = 10.2307/2332142 | jstor = 2332142| year = 1950 | pmid = 15420245 }}</ref><ref>{{cite journal |last1=Li |first1=Hongfei |first2=Catherine A. |last2=Calder|author2-link=Kate Calder |first3=Noel |last3=Cressie|author3-link=Noel Cressie |title=Beyond Moran's ''I'': Testing for Spatial Dependence Based on the Spatial Autoregressive Model |journal=Geographical Analysis |year=2007 |volume=39 |issue=4 |pages=357–375 |doi=10.1111/j.1538-4632.2007.00708.x}}</ref>
* '''[Geary's ''C''](/source/Geary's_C)''' (Geary's Contiguity Ratio): A measure of global spatial autocorrelation developed by [Roy C. Geary](/source/Roy_C._Geary) in 1954.<ref>{{Cite journal  | doi = 10.2307/2986645 | author = [Geary, R. C.](/source/Roy_C._Geary)
 | year = 1954 | title = The Contiguity Ratio and Statistical Mapping | journal = [The Incorporated Statistician](/source/The_Incorporated_Statistician) | volume = 5 | pages = 115&ndash;145 | jstor = 2986645 | issue = 3}}</ref><ref>{{cite journal | doi = 10.2307/2986827 | author = J. N. R. Jeffers | year = 1973 | title = A Basic Subroutine for Geary's Contiguity Ratio | publisher = Wiley | journal = Journal of the Royal Statistical Society, Series D | volume = 22 | issue = 4| pages = 299–302 | jstor = 2986827 }}</ref> It is inversely related to Moran's ''I'', but more sensitive to local autocorrelation than Moran's ''I''.
* '''Getis–Ord ''G''''' (Getis–Ord global G, Geleral G-Statistic): Introduced by [Arthur Getis](/source/Arthur_Getis) and [J. Keith Ord](/source/J._Keith_Ord) in 1992 to supplement Moran's ''I''.<ref>{{Cite journal |last1=Getis |first1=Arthur |last2=Ord |first2=J. Keith |year=1992 |title=The analysis of spatial association by use of distance statistics |journal=Geographical Analysis |volume=24 |issue=3 |pages=189–206 |doi=10.1111/j.1538-4632.1992.tb00261.x}}</ref>

==Local indicators==
thumb|Clusters of the estimated percent of people in poverty by county in the contiguous United States in 2020
thumb|HotSpot map of the estimated percent of non-institutionalized population without health insurance, by county in the contiguous United States in 2020
Notable local indicators of spatial association (LISA) include:<ref name="Grekousis"/>

* '''Local Moran's ''I''''': Derived from Global Moran's ''I'', it was introduced by [Luc Anselin](/source/Luc_Anselin) in 1995<ref>{{Cite journal |last=Anselin |first=Luc |year=1995 |title=Local Indicators of Spatial Association—LISA |journal=Geographical Analysis |volume=27 |issue=2 |pages=93–115 |doi=10.1111/j.1538-4632.1995.tb00338.x|doi-access=free }}</ref> and can be computed using [GeoDa](/source/GeoDa).<ref>{{Cite web |last=Anselin |first=Luc |year=2005 |url=https://www.geos.ed.ac.uk/~gisteac/fspat/geodaworkbook.pdf |title=Exploring Spatial Data with GeoDa: A Workbook |publisher=Spatial Analysis Laboratory |page=138}}</ref>
* '''Getis–Ord ''G<sub>i</sub>''''' (local ''G<sub>i</sub>''): Developed by Getis and Ord based on their global ''G''.
* '''INDICATE's I<sub>N</sub>''': Originally developed to assess the spatial distribution of stars,<ref>{{Cite journal |last1=Buckner |first1=Anne S. M. |last2=Khorrami |first2=Zeinab |last3=Khalaj |first3=Pouria |last4=Lumsden |first4=Stuart L. |last5=Joncour |first5=Isabelle |last6=Moraux |first6=Estelle |last7=Clark |first7=Paul |last8=Oudmaijer |first8=René D. |last9=Blanco |first9=José Manuel |last10=de la Calle |first10=Ignacio |last11=Herrera-Fernandez |first11=José M. |last12=Motte |first12=Frédérique |last13=Salgado |first13=Jesús J. |last14=Valero-Martín |first14=Luis |date=2019-02-01 |title=The spatial evolution of young massive clusters. I. A new tool to quantitatively trace stellar clustering |url=https://ui.adsabs.harvard.edu/abs/2019A&A...622A.184B |journal=Astronomy and Astrophysics |volume=622 |pages=A184 |doi=10.1051/0004-6361/201832936 |arxiv=1901.02371 |bibcode=2019A&A...622A.184B |s2cid=119071236 |issn=0004-6361}}</ref> can be computed for any discrete 2+D dataset using python-based INDICATE tool available from GitHub.<ref>{{Citation |last=abuckner89 |title=abuckner89/INDICATE |date=2021-07-22 |url=https://github.com/abuckner89/INDICATE |access-date=2022-09-14}}</ref>

==See also==
* [Spatial analysis](/source/Spatial_analysis)
* [Tobler's first law of geography](/source/Tobler's_first_law_of_geography)

== References == 
{{Reflist}}

== Further reading ==
* {{Cite journal |last1=Bivand |first1=Roger S. |last2=Wong |first2=David W. S. |year=2018 |title=Comparing implementations of global and local indicators of spatial association |journal=Test |volume=27 |issue=3 |pages=716–748 |url=https://link.springer.com/article/10.1007/s11749-018-0599-x |doi=10.1007/s11749-018-0599-x|hdl=11250/2565494 |s2cid=125895189 |hdl-access=free |url-access=subscription }}

Category:Spatial analysis

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