1. Briggs Henan University 2010
Spatial Autocorrelation: The Single Most Important Concept in Geography and GIS! Introduction to Concepts
Briggs Henan University 2010

2. Briggs Henan University 2010
Spatial Statistics
Descriptive Spatial Statistics: Centrographic Statistics (This time)
single, summary measures of a spatial distribution
- Spatial equivalents of mean, standard deviation, etc..
Inferential Spatial Statistics: Point Pattern Analysis (Next time)
Analysis of point location only--no quantity or magnitude (no attribute variable)
--Quadrat Analysis
--Nearest Neighbor Analysis, Ripley’s K function
Spatial Autocorrelation (Weeks 5 and 6)
One attribute variable with different magnitudes at each location
The Weights Matrix
Global Measures of Spatial Autocorrelation (Moran’s I, Geary’s C, Getis/Ord Global G)
Local Measures of Spatial Autocorrelation (LISA and others)
Prediction with Correlation and Regression (Week 7)
Two or more attribute variables
Standard statistical models
Spatial statistical models  
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3. Briggs Henan University 2010
Point Pattern Analysis (PPA) and Spatial Autocorrelation (SA) : differences and similarities
Point Pattern Analysis (last time)
--points only, and only their location
--there is no “magnitude” value
Spatial Autocorrelation: (this time)
--points and polygons, with different “magnitudes”
-- there is an attribute variable.
--income, rainfall, crime rate, etc.
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4. Spatial Autocorrelation Many ways to define it!
1. The confirmation of Tobler’s first law of geography
Everything is related to everything else, but near things are more related than distant things.
2. Using similarity
The degree to which characteristics at one location are similar (or dissimilar) to those nearby.
3. Using probability
Measure of the extent to which the occurrence of an event in one geographic area makes more probable, or less probable, the occurrence of a similar event in a neighboring geographic area.
4. Using correlation
Correlation of a variable with itself through space.
The correlation between an observation’s value on a variable and the value of near-by observations on the same variable
Lets look at these in more detail.

5. Spatial Autocorrelation: 1. Tobler’s Law
The confirmation of Tobler’s first law of geography*:
Everything is related to everything else, but near things are more related than distant things.
The single most important concept in geography and GIS!
*Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography, 46(2):
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6. Briggs Henan University 2010
Positive Spatial Autocorrelation
Spatial
Autocorrelation
Spatial:
On a map
Auto:
Self
Correlation:
Degree of
relative
similarity
Positive: similar values cluster together on a map
Negative Spatial Autocorrelation
Source: Dr Dan Griffith, with modification
Negative: dissimilar (different) values cluster together on a map
Briggs Henan University 2010

7. Positive spatial autocorrelation - high values
surrounded by nearby high values
- intermediate values surrounded
by nearby intermediate values
- low values surrounded by
nearby low values
2002 population
density

8. Negative spatial autocorrelation - high values
surrounded by nearby low values
- intermediate values surrounded
by nearby intermediate values
- low values surrounded by
nearby high values
competition for space
Grocery store density

9. Spatial Autocorrelation: more ways to describe it
2. Based on Similarity
The degree to which characteristics at one location are similar to (or different from) those nearby.
Similar to = positive spatial autocorrelation
Different from (dissimilar) = negative spatial autocorrelation
Positive spatial autocorrelation much more common than negative
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10. Spatial Autocorrelation Exists Everywhere!
POLLUTION MONITORING
SATELLITE IMAGE
HOUSEHOLD SAMPLING
AGRICULTURAL EXPERIMENT
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11. Spatial Autocorrelation: more ways to describe it
Based on Probability
Measure of the extent to which the occurrence of an event in one geographic unit (polygon) makes more probable, or less probable, the occurrence of a similar event in a neighboring unit.
Do you recognize this from earlier discussion?
It’s the same concept as clustered, random, dispersed!
high negative spatial autocorrelation
no spatial autocorrelation*
high positive spatial autocorrelation
Dispersed Pattern
Random Pattern
Clustered Pattern
CLUSTERED
DISPERSED
UNIFORM/
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12. Even More Ways to Describe SA
4. Using correlation
Correlation of a variable with itself through space.
The correlation between an observation’s value on a variable and the value of near-by observations on the same variable.
Correlation = “similarity”, “association”, or “relationship”
Scatter diagram
Crime rate in
near-by area
Crime rate in an area
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13. Scatter Diagram: how is it different?
Standard Statistics: shows the association or relationship between two different variables
Spatial Autocorrelation: shows the association or relationship between the same variable in “near-by” areas.
Each point is a
geographic location
Education
“next door”
income
In a neighboring or near-b y area
education
education
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14. Why is Spatial Autocorrelation Important?
Two reasons
Spatial autocorrelation is important because it implies the existence of a spatial process
Why are near-by areas similar to each other?
Why do high income people live “next door” to each other?
These are GEOGRAPHICAL questions.
They are about location
2. It invalidates most traditional statistical inference tests
If SA exists, then the results of standard statistical inference tests may be incorrect (wrong!)
We need to use spatial statistical inference tests
Create
Processes
Pattern
Population
Infer
Sample
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15. Why are standard statistical tests wrong?
Statistical tests are based on the assumption that the values of observations in each sample are independent of one another
spatial autocorrelation violates this
samples taken from nearby areas are related to each other and are not independent
Implies a relationship between nearby observations
Values near each other are similar in magnitude.
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16. Briggs Henan University 2010
Why are standard statistical tests wrong? Example for the correlation coefficient (r)
What is the correlation coefficient (r)?
The most common statistic in all of science
measures the strength of the relationship (or “association”) between two variables e.g. income and education
Varies on a scale from –1 thru 0 to +1
+1 implies a perfect positive association
As values go up () on one, they also go up () on the other
income and education
0 implies no association
-1 implies perfect negative association
As values go up on one () , they go down () on the other
price and quantity purchased
Full name is the Pearson Product Moment correlation coefficient, -1 +1
() ()
() ()
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17. Examples of Scatter Diagrams and the Correlation Coefficient
Positive
Education
Income
r = 0.26
weak
positive
r = 1
r = 0.72
perfect
positive
strong
positive
Negative
r = -0.71
strong
negative
Price
Quantity
r = -1
perfect
negative
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18. Briggs Henan University 2010
Why are standard statistical tests wrong? Example for the correlation coefficient (r)
If Spatial Autocorrelation exists:
Correlation coefficients appear to be bigger than they really are, and
They are more likely to be found “statistically significant”
You are “fooled twice”:
--you are more likely to incorrectly conclude
a relationship exists when it does not
--You believe that the relationship is
stronger than it really is
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19. Briggs Henan University 2010
Why are standard statistical tests wrong? Example for the correlation coefficient (r)
If Spatial Autocorrelation exists:
Correlation coefficients bigger than they really are
because income and
education are similar in
near by areas
Correlation coefficient is
“biased upward”
Also, more likely to appear “statistically significant”
standard error is smaller because spatial autocorrelation “artificially” reduces variability
there is actually more variability than it appears
“exagerated precision”
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20. Measuring Spatial Autocorrelation:
the problem of measuring
“nearness” or “proximity”
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21. Measuring Spatial Autocorrelation: the problem of measuring “nearness”
To measure spatial autocorrelation, we must know the “nearness” of our observations
Which points or polygons are “ near” or “next to” other points or polygons?
Which provinces are near Henan?
How measure this?
Seems simple and obvious,
but it is not!
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22. Measuring Spatial Autocorrelation: the Spatial Weights matrix
Wij the spatial weights matrix measures the relative location of all points i and j,
Different methods of calculating Wij can result in different values for autocorrelation and different conclusions from statistical significance tests!
Wij
Admin_Name
Anhui
Beijing
Chongqing
Fujian
Gansu
Guangdong
Guangxi
Guizhou
Hainan
Hebei
Heilongjiang
Henan
…….
Zhejiang
……… ?
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23. Measuring Relative Spatial Location: Contiguity and Distance
Two methods for measuring nearness
1. Weights based on Contiguity--binary (0,1)
If zone j is next to zone i, it receives a weight of 1
otherwise it receives a weight of 0,
It is essentially excluded
But what constitutes contiguity? Not as easy as it seems!
2. Weights based on Distance—continuous variable
Measure the actual distance between points, or between polygon centroids
But what measure do we use, and
distance to what points -- All? Some?
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24. Spatial neighbors based on contiguity* (adjacency)
* Shares common border
Sharing a border or boundary
Rook: sharing a border
Queen: sharing a border or a point
rook
queen
Hexagons
Irregular
Which use?
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25. Spatial weights matrix for Rook case
Matrix contains a:
1 if share a border
0 if do not share a border
associated
geographic connectivity/
weights matrix
4 areal units
4x4 matrix A B C D 1 A B C D
W =
Common border
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26. Problem Situations for Irregular Polygons Many!
“Close” but no common border
Include polygons which have a centroid
within the “convex hull” for the centroids of polygons that do share a common border
Length of border
Is Shanxi “as close to” Nei Mongol as to Henan?
Base “closeness” on proportion of shared border,
not just one (1) or zero (0)
wij = border lengthij /border lengthj) X
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27. Briggs Henan University 2010
Measuring Contiguity: Lagged Contiguity Should we include second order contiguity?
1st
order
Nearest neighbor
rook
hexagon
queen
2nd
order
Next nearest neighbor
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28. Formats for Weights Matrix
Raw versus row standardized
Full contiguity versus sparse contiguity
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29. Row-standardized geographic contiguity matrices A B C D E F
Divide each number by the row sum
Total number of neighbors
--some have more than others
Row standardized
--usually use this A B C D E F
Row Sum
0.0
0.5 1
0.3 A B C D E F
Row Sum 1 2 3
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30. Briggs Henan University 2010
Queens Case Full Contiguity Matrix for US States
Column headings not shown (same as rows)
Principal diagonal has 0s (blanks)
other 0s omitted for simplicity
Can be very large, thus inefficient to use.
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31. Briggs Henan University 2010
Queens Case Sparse Contiguity Matrix for US States
Ncount is the number of neighbors for each state
Max is 8 (Missouri and Tennessee)
Sum of Ncount is 218
Number of common borders (joins)
ncount / 2 = 109
N1, N2… FIPS codes for neighbors
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32. Briggs Henan University 2010
Challenge for You
Which China province has the most neighbors?
How many does it have?
Create contiguity matrices for the Provinces of China
Can be done with GeoDA or with ArcGIS
Or you can do it “by hand”
Use the software to see if you get it correct
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33. Weights Based on Distance again, not that simple
Functional Form to use?
Distance metric to use ?
Which points/polygons to include?
How measure distance between polygons?
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34. Weights Based on Distance
1. Functional Form
We want “nearness” not distance
Most common choice is the inverse (reciprocal) of the distance between locations i and j (wij = 1/dij)
Other functions also used
inverse of squared distance (wij =1/dij2), or
negative exponential (wij = e-d or wij = e-d2)
nearness
distance
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35. Weights Based on Distance
2. Distance metric
2-D Cartesian distance via Pythagorus
Use for projected data
3-D Spherical distance via spherical coordinates
Cos d = (sin a sin b) + (cos a cos b cos P)
where: d = arc distance
a = Latitude of A
b = Latitude of B
P = degrees of long. A to B
Use for unprojected data
possible distance metrics:
Euclidean straight line/airline
city block/manhattan metric
distance through network
Appropriate if within a city
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36. Weights based on Distance
3. What points/polygons to include?
Distances to all points/polygons?
If use all, may make it impossible to solve necessary equations: matrix too big
May not make theoretical sense: effects may only be ‘local’
Is Henan influenced by Xinjiang?
Include distance to only the “nth” nearest neighbors
How many is n? First? Second?
Include distances to locations only within a buffer distance

37. Weights based on Distance
4. Measuring distance between polygons
distances usually measured centroid to centroid, but
could be measured from boundary of one polygon to centroid of others
could be measured between the two closest boundary points
adjustment required for contiguous polygons since distance for these would be zero
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38. Many decisions! Many challenges!
That is what makes research fun!
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39. What have we learned today?
The concept of spatial autocorrelation.
“Near things are more similar than distant things”
The use of the weights matrix Wij to measure “nearness”
The difficulty of measuring “nearness”
That is a surprise!
Next Time
Measures of Spatial Autocorrelation
Join Count statistic Geary’s C
Moran’s I Getis-Ord G statistic
Briggs Henan University 2010

40. Briggs Henan University 2010
Challenge for You
Which China province has the most neighbors?
How many does it have?
Create contiguity matrices for the Provinces of China
Can be done with GeoDA or with ArcGIS
Or you can do it “by hand”
Use the software to see if you get it correct
Briggs Henan University 2010

41. Appendix: A Note on Sampling Assumptions
Another factor which influences results from these tests is the assumption made regarding the type of sampling involved:
Free (or normality) sampling
Analogous to sampling with replacement
After a polygon is selected for a sample, it is returned to the population set
The same polygon can occur more than one time in a sample
Non-free (or randomization) sampling
Analogous to sampling without replacement
After a polygon is selected for a sample, it is not returned to the population set
The same polygon can occur only one time in a sample
The formulae used to calculate test statistics (particularly the standard error) differ depending on which assumption is made
Generally, the formulae are substantially more complex for randomization sampling—unfortunately, it is also the more common situation!
Usually, assuming normality sampling requires knowledge about larger trends from outside the region or access to additional information within the region in order to estimate parameters.
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42. Briggs Henan University 2010