1. Local Measures of Spatial Autocorrelation
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2. 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
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
China
An equivalent local measure can be calculated for most global measures
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3. 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:
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4. 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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5. Local Indicators of Spatial Association (LISA)
Raw data
Since a measure is available for each polygon, these can be mapped to indicate how spatial autocorrelation varies over the study region
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
LISA
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6. Calculating Anselin’s LISA
The local Moran statistic for areal unit i is:
where zi is the original variable xi in
“standardized form”
or it can be in “deviation form”
and wij is the spatial weight
The summation is across each row i of the spatial weights matrix.
An example follows
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8. Briggs Henan University 2010
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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9. Briggs Henan University 2010
Contiguity Matrix 1 2 3 4 5 6 7
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
Sum
Neighbors
Illiteracy
14.49
9.36
6 2 1
6.49
7 2 1
8.05
6 1
7.36
1 3 5
7.69
2 4
3.97 1 5 4 3 6 7 2
Each row in the contiguity matrix describes the neighborhood for that location.
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10. Contiguity Matrix and Row Standardized Spatial Weights Matrix1 2 3 4 5 6 7
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai
Sum
Row Standardized Spatial Weights Matrix
0.00
0.20
0.25
0.33
0.50
1/3
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11. Calculating standardized (z) scores
Deviations from Mean and z scores. X
X-Xmean
X-Mean2 z
Anhui
14.49
6.29
39.55
2.101
Zhejiang
9.36
1.16
1.34
0.387
Jiangxi
6.49
(1.71)
2.93
(0.572)
Jiangsu
8.05
(0.15)
0.02
(0.051)
Henan
7.36
(0.84)
0.71
(0.281)
Hubei
7.69
(0.51)
0.26
(0.171)
Shanghai
3.97
(4.23)
17.90
(1.414)
Mean and Standard Deviation
Sum
57.41
0.00
62.71
Mean
/ =
8.20
Variance
/ =
8.96 SD
√ 8.96 =
2.99
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12. Calculating LISA wij zj wijzj Anhui Zhejiang Jiangxi Jiangsu Henan
Row Standardized Spatial Weights Matrix
Code
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai 1
0.00
0.20 2
0.25 3
0.33 4 5
0.50 6 7
wij
Z-Scores for row Province and its potential neighbors
Anhui
Zhejiang
Jiangxi
Jiangsu
Henan
Hubei
Shanghai Zi
2.101
0.387
(0.572)
(0.051)
(0.281)
(0.171)
(1.414)
Spatial Weight Matrix multiplied by Z-Score Matrix (cell by cell multiplication)
SumWijZj
LISA
Lisa from
0.000
GeoDA -
0.077
(0.114)
(0.010)
(0.056)
(0.034)
(0.137)
-0.289
-0.248
0.525
(0.143)
(0.013)
(0.353)
0.016
0.006
0.005
0.700
0.129
(0.057)
0.772
-0.442
-0.379
(0.471)
0.358
-0.018
-0.016
1.050
(0.085)
0.965
-0.271
-0.233
(0.191)
(0.094)
0.416
-0.071
-0.061
0.194
(0.025)
0.168
-0.238
-0.204 zj
wijzj

13. Results Raw Data Moran’s I = -.01889 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
14.49
-0.25
0.12
Zhejiang
9.36
0.01
0.46
Jiangxi
6.49
-0.38
0.04
Jiangsu
8.05
-0.02
0.32
Henan
7.36
-0.23
0.14
Hubei
7.69
-0.06
0.28
Shanghai
3.97
-0.20
0.37
Low-High

14. LISA for Illiteracy for all China Provinces
Low
High
Illiteracy Rates
LISA
Moran’s I =
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15. Briggs Henan University 2010
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
High/High positive SA
Low/Low positive SA
High/Low negative SA
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16. 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
Locations of positive spatial association
(“I’m similar to my neighbors”). WX Q2 Q1
Q1 (values [+], nearby values [+]): H-H
Q3 (values [-], nearby values [-]): L-L
Locations of negative spatial association
(“I’m different from my neighbors”). Q3 Q4 X
Low/Low positive SA
High/Low negative SA
Q2 (values [-], nearby values [+]): L-H
Q4 (values [+], nearby values [-]): H-L
GISC 7361 Spatial Statistics

17. Why is Moran’s I low for China provinces?
For illiteracy = .2047
Are provinces really “local”
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18. Briggs Henan University 2010
LISA for Median Income, 2000 in D/FW
Source: Eric Hajek, 2008
Moran’s I = .59
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19. Examples of LISA for 7 Ohio counties: median income
Lake
Ashtabula
Geauga
Cuyahoga
Trumbull
Summit
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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20. Local Getis-Ord G Statistic
It is the proportion of all x values in the study area accounted for by the neighbors of location i
Local Moran’s I
Global
Getis-Ord G
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.
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21. LISA and Getis G with Different Distance Weights
Getis with
0.5 distance
Data for crime in Columbus, Ohio from Anselin, 1995
LISA with
0,1 contiguity
Results for Getis G vary depending on distance band used.
Getis with
1.0 distance
Getis with
2.0 distance

22. Briggs Henan University 2010
Running in ArcGIS
Local Getis G*
with Fixed Distance Bands at
2, 1, and 0.5
LISA using Contiguity Weights
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23. Briggs Henan University 2010
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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24. Briggs Henan University 2010
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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25. 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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26. 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
Gentrification?
High value/
High crime
Unique:
Low value/
Low crime
Classic suburb:
high value/
low crime
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27. 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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28. Briggs Henan University 2010
References
Getis, A. and Ord, J.K. (1992) The analysis of spatial association by use of distance statistics Geographical Analysis, 24(3)
Ord, J.K. and Getis A. (1995) Local Spatial Autocorrelation Statistics: distributional issues and an application Geographical Analysis, 27(4)
Anselin, L. (1995) Local Indicators of Spatial Association-LISA Geographical Analysis 27:
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