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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Spatial autocorrelation statistic
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 [ edit ]
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
i
{\displaystyle i}
is included or not[6]
G
i
=
∑
j
≠
i
w
i
j
x
j
∑
j
≠
i
x
j
{\displaystyle G_{i}={\frac {\sum _{j\neq i}w_{ij}x_{j}}{\sum _{j\neq i}x_{j}}}}
G
i
∗
=
∑
j
w
i
j
x
j
∑
j
x
j
{\displaystyle G_{i}^{*}={\frac {\sum _{j}w_{ij}x_{j}}{\sum _{j}x_{j}}}}
Here
x
i
{\displaystyle x_{i}}
is the value observed at the
i
t
h
{\displaystyle i^{th}}
spatial site and
w
i
j
{\displaystyle w_{ij}}
is the spatial weight matrix which constrains which sites are connected to one another. For
G
i
∗
{\displaystyle 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 .[6]
Global statistics [ edit ]
The Getis-Ord statistics of overall spatial association are[7] [9]
G
=
∑
i
j
,
i
≠
j
w
i
j
x
i
x
j
∑
i
j
,
i
≠
j
x
i
x
j
{\displaystyle G={\frac {\sum _{ij,i\neq j}w_{ij}x_{i}x_{j}}{\sum _{ij,i\neq j}x_{i}x_{j}}}}
G
∗
=
∑
i
j
w
i
j
x
i
x
j
∑
i
j
x
i
x
j
{\displaystyle G^{*}={\frac {\sum _{ij}w_{ij}x_{i}x_{j}}{\sum _{ij}x_{i}x_{j}}}}
The local and global
G
∗
{\displaystyle G^{*}}
statistics are related through the weighted average
∑
i
x
i
G
i
∗
∑
i
x
i
=
∑
i
j
x
i
w
i
j
x
j
∑
i
x
i
∑
j
x
j
=
G
∗
{\displaystyle {\frac {\sum _{i}x_{i}G_{i}^{*}}{\sum _{i}x_{i}}}={\frac {\sum _{ij}x_{i}w_{ij}x_{j}}{\sum _{i}x_{i}\sum _{j}x_{j}}}=G^{*}}
The relationship of the
G
{\displaystyle G}
and
G
i
{\displaystyle G_{i}}
statistics is more complicated due to the dependence of the denominator of
G
i
{\displaystyle G_{i}}
on
i
{\displaystyle i}
.
Relation to Moran's I [ edit ]
Moran's I is another commonly used measure of spatial association defined by
I
=
N
W
∑
i
j
w
i
j
(
x
i
−
x
¯
)
(
x
j
−
x
¯
)
∑
i
(
x
i
−
x
¯
)
2
{\displaystyle 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
{\displaystyle N}
is the number of spatial sites and
W
=
∑
i
j
w
i
j
{\displaystyle W=\sum _{ij}w_{ij}}
is the sum of the entries in the spatial weight matrix. Getis and Ord show[7] that
I
=
(
K
1
/
K
2
)
G
−
K
2
x
¯
∑
i
(
w
i
⋅
+
w
⋅
i
)
x
i
+
K
2
x
¯
2
W
{\displaystyle 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
⋅
=
∑
j
w
i
j
{\displaystyle w_{i\cdot }=\sum _{j}w_{ij}}
,
w
⋅
i
=
∑
j
w
j
i
{\displaystyle w_{\cdot i}=\sum _{j}w_{ji}}
,
K
1
=
(
∑
i
j
,
i
≠
j
x
i
x
j
)
−
1
{\displaystyle K_{1}=\left(\sum _{ij,i\neq j}x_{i}x_{j}\right)^{-1}}
and
K
2
=
W
N
(
∑
i
(
x
i
−
x
¯
)
2
)
−
1
{\displaystyle K_{2}={\frac {W}{N}}\left(\sum _{i}(x_{i}-{\bar {x}})^{2}\right)^{-1}}
. They are equal if
w
i
j
=
w
{\displaystyle w_{ij}=w}
is constant, but not in general.
Ord and Getis[8] also show that Moran's I can be written in terms of
G
i
∗
{\displaystyle G_{i}^{*}}
I
=
1
W
(
∑
i
z
i
V
i
G
i
∗
−
N
)
{\displaystyle I={\frac {1}{W}}\left(\sum _{i}z_{i}V_{i}G_{i}^{*}-N\right)}
where
z
i
=
(
x
i
−
x
¯
)
/
s
{\displaystyle z_{i}=(x_{i}-{\bar {x}})/s}
,
s
{\displaystyle s}
is the standard deviation of
x
{\displaystyle x}
and
V
i
2
=
1
N
−
1
∑
j
(
w
i
j
−
1
N
∑
k
w
i
k
)
2
{\displaystyle 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
i
j
{\displaystyle w_{ij}}
.
See also [ edit ]
References [ edit ]
^ https://pysal.org/
^ "R-spatial/Spdep" . GitHub .
^ 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 .
^ a b "Local Spatial Autocorrelation (2 )" .
^ 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 .
^ 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 .
^ "How High/Low Clustering (Getis-Ord General G) works—ArcGIS Pro | Documentation" .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Getis–Ord_statistics&oldid=1222989590 "
C a t e g o r i e s :
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● T h i s p a g e w a s l a s t e d i t e d o n 9 M a y 2 0 2 4 , a t 0 5 : 2 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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