Local Measures of
Spatial Autocorrelation
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Global Measures and Local Measures
• Global Measures (last time)
– A single value which applies to the entire data set
• The same pattern or process occurs over the entire
geographic area
China
• An average for the entire area
• Local Measures (this time)
– A value calculated for each observation unit
• Different patterns or processes may occur in different
parts of the region
• A unique number for each location
An equivalent local measure can be
calculated for most global measures
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Local Indicators of Spatial Association (LISA)
• We will look at local versions of Moran’s I, Geary’s C,
and the Getis-Ord G statistic
• Moran’s I is most commonly used, and the local version
is often called Anselin’s LISA, or just LISA
See:
Luc Anselin 1995 Local Indicators of Spatial
Association-LISA Geographical Analysis 27: 93-115
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Local Indicators of Spatial Association (LISA)
– The statistic is calculated for each
areal unit in the data
– For each polygon, the index is
calculated based on neighboring
polygons with which it shares a border
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Local Indicators of Spatial Association (LISA)
– Since a measure is
available for each polygon,
these can be mapped to
indicate how spatial
autocorrelation varies over
the study region
Raw data
LISA
– Since each index has an
associated test statistic, we
can also map which of the
polygons has a statistically
significant relationship
with its neighbors, and
show type of relationship
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Calculating Anselin’s LISA
• The local Moran statistic for areal unit i is:
I i  zi  wij z j
j
where zi is the original variable xi in z  xi  x
i
SDx
“standardized form”
or it can be in “deviation form” x  x
i
and wij is the spatial weight
The summation j is across each row i of the
spatial weights matrix.
An example follows
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Example using seven China provinces
--caution: “edge effects” will strongly influences the
results because we have a very small number of
observations
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Contiguity Matrix
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
1
2
3
4
5
6
7
1
2
3
4
5
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
0
1
1
1
1
1
0
1
0
1
1
0
0
1
1
1
0
0
0
1
0
1
1
0
0
0
0
1
1
0
0
0
0
1
0
6
7
Hubei Shanghai
1
0
1
0
1
0
0
0
1
0
1
0
0
0
Sum
5
4
3
3
2
3
2
5
Each row in the contiguity matrix
describes the neighborhood for
that location.
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Neighbors
Illiteracy
14.49
9.36
6.49
8.05
7.36
7.69
3.97
65432
7431
621
721
61
135
24
4
1
7
2
3
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Contiguity Matrix and
Row Standardized Spatial Weights Matrix
Contiguity Matrix
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
1
2
3
4
5
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
1
2
3
4
5
6
0
1
1
1
1
1
1
0
1
1
0
0
1
1
0
0
0
1
1
1
0
0
0
0
1
0
0
0
0
1
1
0
1
0
1
0
0
1
0
1
0
0
5
4
3
3
2
3
7
0
1
0
1
0
0
0
2
Jiangxi
Jiangsu
Henan
Row Standardized Spatial Weights Matrix
Anhui
Zhejiang
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
6
7
Hubei Shanghai
Hubei Shanghai
Sum
Sum
1
2
3
4
5
6
0.00
0.25
0.33
0.33
0.50
0.33
0.20
0.00
0.33
0.33
0.00
0.00
0.20
0.25
0.00
0.00
0.00
0.33
0.20
0.25
0.00
0.00
0.00
0.00
0.20
0.00
0.00
0.00
0.00
0.33
0.20
0.00
0.33
0.00
0.50
0.00
0.00
0.25
0.00
0.33
0.00
0.00
1
1
1
1
1
1
7
0.00
0.50
0.00
0.50
0.00
0.00
0.00
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Calculating standardized (z) scores
Deviations from Mean and z scores.
X
X-Xmean
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
14.49
9.36
6.49
8.05
7.36
7.69
3.97
X-Mean2
z
6.29
39.55
2.101
1.16
1.34
0.387
(1.71)
2.93
(0.572)
(0.15)
0.02
(0.051)
(0.84)
0.71
(0.281)
(0.51)
0.26
(0.171)
(4.23)
17.90
(1.414)
Mean and Standard Deviation
Sum
57.41
0.00
Mean
57.41 / 7 =
Variance
62.71 / 7 =
SD
√ 8.96
=
xi  x
zi 
SDx
62.71
8.20
8.96
2.99
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Calculating LISA
Row Standardized Spatial Weights
Matrix
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
1
0.00
0.20
0.20
0.20
0.20
0.20
0.00
2
0.25
0.00
0.25
0.25
0.00
0.00
0.25
3
0.33
0.33
0.00
0.00
0.00
0.33
0.00
4
0.33
0.33
0.00
0.00
0.00
0.00
0.33
5
0.50
0.00
0.00
0.00
0.00
0.50
0.00
6
0.33
0.00
0.33
0.00
0.33
0.00
0.00
7
0.00
0.50
0.00
0.50
0.00
0.00
0.00
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
Hubei Shanghai
wij
I i  zi  wij z j
Z-Scores for row Province and its potential neighbors
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
Zi
Anhui
2.101
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Zhejiang
0.387
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Jiangxi
(0.572)
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Jiangsu
(0.051)
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Henan
Hubei
(0.281)
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
(0.171)
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Shanghai
(1.414)
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Zhejiang
Jiangxi
Jiangsu
Henan
zj
wijzj
Spatial Weight Matrix multiplied by Z-Score Matrix (cell by cell multiplication)
Anhui
j
Hubei
Shanghai SumWijZj
Zi
LISA
0.000
Lisa from
GeoDA
Anhui
2.101
-
0.077
(0.114)
(0.010)
(0.056)
(0.034)
-
(0.137)
-0.289
-0.248
Zhejiang
0.387
0.525
-
(0.143)
(0.013)
-
-
(0.353)
0.016
0.006
0.005
Jiangxi
(0.572)
0.700
0.129
-
-
-
(0.057)
-
0.772
-0.442
-0.379
Jiangsu
(0.051)
0.700
0.129
-
-
-
-
(0.471)
0.358
-0.018
-0.016
Henan
(0.281)
1.050
-
-
-
-
(0.085)
-
0.965
-0.271
-0.233
Hubei
(0.171)
0.700
-
(0.191)
-
(0.094)
-
-
0.416
-0.071
-0.061
Shanghai
(1.414)
-
0.194
-
(0.025)
-
-
-
0.168
-0.238
-0.204
Results
Moran’s I = -.01889
Raw Data
I expected Anhui to be
High-Low!
(high illiteracy
surrounded by low)
Low
Significance levels are calculated by
simulations. They may differ each
time software is run.
High
Province
Literacy %
LISA
Significance
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
14.49
9.36
6.49
8.05
7.36
7.69
3.97
-0.25
0.12
0.01
0.46
-0.38
0.04
-0.02
0.32
-0.23
0.14
-0.06
0.28
-0.20
0.37
Low-High
LISA for Illiteracy for all China Provinces
Low
High
Illiteracy Rates
LISA
Moran’s I = 0.2047
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Moran Scatter Plot
Scatter Diagram between X and Lag-X, the “spatial lag” of X formed
by averaging all the values of X for the neighboring polygons
Identifies which type of spatial autocorrelation exists.
Low/High
negative SA
Low/Low
positive SA
High/High
positive SA
High/Low
negative SA
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Quadrants of Moran Scatterplot
Each quadrant corresponds to one of the four
different types of spatial association (SA)
Low/High
negative SA
High/High
positive SA
WX
Q2
Q1
Locations of positive spatial association
(“I’m similar to my neighbors”).
Q1 (values [+], nearby values [+]): H-H
Q3 (values [-], nearby values [-]): L-L
0
Q3
Low/Low
positive SA
Locations of negative spatial association
(“I’m different from my neighbors”).
Q4
0
X
High/Low
Q2 (values [-], nearby values [+]): L-H
negative SA
Q4 (values [+], nearby values [-]): H-L
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GISC 7361 Spatial Statistics
Why is Moran’s I low for China provinces?
• For illiteracy = .2047
• Are provinces really “local”
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LISA for Median Income, 2000 in D/FW
Source: Eric Hajek, 2008
Moran’s I = .59
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Examples of LISA for 7 Ohio counties: median income
Ashtabula
Lake
Geauga
Cuyahoga
Summit
Trumbull
Portage
Ashtabula has a statistically significant
Negative spatial autocorrelation ‘cos it is
a poor county surrounded by rich ones
(Geauga and Lake in particular)
Source: Lee and Wong
Median
Income
(p< 0.10)
(p< 0.05)
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Local Getis-Ord G Statistic
w x
G (d ) 
x
ij
j
j
i
j
j
I i  zi  wij z j Local
j
Local Getis-Ord
It is the proportion of all x values
in the study area accounted for by
the neighbors of location i
Moran’s I
  w (d )x x Global
G (d ) 
  x x Getis-Ord G
ij
i
For comparison
G will be high where high values cluster
G will be low where low values cluster
Interpreted relative to expected value
if randomly distributed.
i
j
j
i
i
j
j
 w (d )
ij
E (Gi (d )) 
j
n 1
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LISA and Getis G with Different Distance Weights
LISA with
0,1 contiguity
Data for crime
in Columbus,
Ohio from
Anselin, 1995
Getis with
0.5 distance
Results for Getis G
vary depending on
distance band used.
Getis with
1.0 distance
Getis with
2.0 distance
Running in ArcGIS
Local Getis G*
with Fixed Distance Bands at
2, 1, and 0.5
LISA using Contiguity Weights
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Can map the statistical
significance level and
use it as a measure of
the “strength” of the
spatial autocorrelation
--note how the
significance level is
higher at the center of
each cluster.
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Bivariate LISA
Moran Scatter Plot for Crime v. Income
• Moran’s I is the correlation between X
and Lag-X--the same variable but in
nearby areas
– Univariate Moran’s I
• Bivariate Moran’s I is a correlation
between X and a different variable in
nearby areas.
Moran Significance Map for Crime v. Income Moran Cluster Map for Crime v. Income
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Bivariate LISA
and the Correlation
Coefficient
Scatter Diagram for relationship
between income and crime
Correlation coefficient r = 0.696
• Correlation Coefficient is
the relationship between
two different variables in
the same area
• Bivariate LISA is a
correlation between two
different variables in an
area and in nearby areas.
Bivariate Moran’s I:
--less strong relationship
--greater scatter
--lower slope
Moran’s I = -.45
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Bivariate LISA:
a local version of the correlation coefficient
– Can view Bivariate LISA as a “local” version of the correlation coefficient
– It shows how the nature & strength of the association between two variables
varies over the study region
– For example, how home values are associated with crime in surrounding areas
Classic Inner City:
Low value/
High crime
Unique:
Low value/
Low crime
Gentrification?
High value/
High crime
Classic suburb:
high value/
low crime
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What have we learned today?
Local Indicators of Spatial Autocorrelation
– Anselin’s LISA
– Local Getis Ord G
Spatial autocorrelation can be calculated for each areal unit
Spatial autocorrelation can vary across the region in strength
and in type
Next time (Friday)
Using GeoDA software to explore spatial
autocorrelation
Next week
Spatial regression and modeling
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References
• Getis, A. and Ord, J.K. (1992) The analysis of
spatial association by use of distance statistics
Geographical Analysis, 24(3) 189-206
• Ord, J.K. and Getis A. (1995) Local Spatial
Autocorrelation Statistics: distributional issues and
an application Geographical Analysis, 27(4) 286306
• Anselin, L. (1995) Local Indicators of Spatial
Association-LISA Geographical Analysis 27: 93115
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