1. Spatial Autocorrelation using GIS
Jennie Murack

2. Objectives Understand the concept of spatial autocorrelation
Learn which tools to use in Geoda and Arcmap to test for autocorrelation
Interpret output from spatial autocorrelation tests

3. What is spatial autocorrelation?

4. What is spatial autocorrelation?
Based on Tobler’s first law of geography, “Everything is related to everything else, but near things are more related than distant things.”
It’s the correlation of a variable with itself through space.
Patterns may indicate that data are not independent of one another, violating the assumption of independence for some statistical tests.

5. Tests for spatial autocorrelation will allow you to answer the following questions about your data:
How are the features distributed?
What is the pattern created by the features?
Where are the clusters?
How do patterns and clusters of different variables compare to one another?

6. Patterns
Useful to:
Better understand geographic phenomena (ex. Habitats)
Monitor conditions (ex. Level of clustering)
Compare different sets of features (ex. Patterns of different types of crimes)
Track change

7. Patterns
You can measure the pattern formed by the location of features or patterns of attribute values associated with features (ex. median home value, percent female, etc.).
New AIDS cases in 1994
New AIDS cases in 2003

8. Types of data often analyzed
Location of crimes, animals, retail, industry, etc.
Land cover
Land use
Census/social data

9. Software ArcGIS GeoDa – open source
Complete GIS software with hundreds of tools
Can work with several datasets (layers) at once.
GeoDa – open source
Solely for spatial statistics
Use one dataset (layer) at a time.
Simple, easy-to-use, interface
Available with registration at:

10. Conceptual Models

11. Spatial Neighborhoods and Weights
Neighborhood = area in which the GIS will compare the target values to neighboring values
Neighborhoods are most often defined based on adjacency or distance, but can be defined based on travel time, travel cost, etc.
You can also define a cutoff distance, the amount of adjacency (borders vs. corners), or the amount of influence at different distances
A table of spatial weights is used to incorporate these definitions into statistical analysis.

12. Distance Models
Inverse distance – all features influence all other features, but the closer something is, the more influence it has
Distance band – features outside a specified distance do not influence the features within the area
Zone of indifference – combines inverse distance and distance band
Distance could be Euclidean or Manhattan
You choose the model for certain calculations based on what you know about the data.

13. Adjacency Models
K Nearest Neighbors – a specified number of neighboring features are included in calculations
Polygon Contiguity – polygons that share an edge or node influence each other
Spatial weights – specified by user (ex. Travel times or distances)

14. Types of Contiguity Rook = Share edges Bishop = share corners
Queen = share edges or corners
Secondary order contiguity = neighbor of neighbor
Image from:

15. Measuring Data Locations

16. Average Nearest Neighbor
Measures how similar the actual mean distance is to the expected mean distance for a random distribution
Measures clustering vs. dispersion of feature locations
Can be used to compare distributions to one another
Concerns: one point on a line is chosen for analysis, extent of study area can affect results (many features near the edge of the study bias results)
Z score used to determine significance
Features at an edge bias the study because they do not have as many neighbors

17. Ripley’s K-function
GIS counts the number of neighboring features within a given distance to each feature.
Like Nearest Neighbor, the K-function measures clustering/dispersion of feature locations, but includes neighbors occurring within a certain distance.
It is often used with individual points.
The test compares the observed K value at each distance to the expected K value for a random distribution at each distance.
Concerns: points at the edge of the study area may have few neighbors

18. Ripley’s K-function
We won’t be using these in the exercises, but they are useful to know they exist.
Assaults are clustered until about 13,000 ft. and then dispersed beyond 15,000 ft.

19. Measuring Data Values

20. Global vs. Local Statistics
Global statistics – identify and measure the pattern of the entire study area
Do not indicate where specific patterns occur
Local Statistics – identify variation across the study area, focusing on individual features and their relationships to nearby features (i.e. specific areas of clustering)

21. Spatial Autocorrelation (Moran’s I)
Global statistic
Measures whether the pattern of feature values is clustered, dispersed, or random.
Compares the difference between the mean of the target feature and the mean for all features to the difference between the mean for each neighbor and the mean for all features.
For more information on the equation, see ESRI online help
Mean of Target Feature
Mean of each neighbor
Mean of all features

22. Spatial Autocorrelation (Moran’s I)
Calculates I values:
I=0=random distribution
I<0=Values dispersed
0<I=Values clustered
Clustered
Dispersed
Random

23. Spatial Autocorrelation (Moran’s I)
I= -.12, slightly dispersed
Moran’s I shows the similarity of nearby features through the I value (-1 to 1), but does not indicate if the clustering is for high values or low values.
I= .26, clustered

24. Anselin Local Moran’s I
Local statistic
Measures the strength of patterns for each specific feature.
Compares the value of each feature in a pair to the mean value for all features in the study area.
See ESRI online help for the equation.

25. Anselin Local Moran’s I
Positive I value:
Feature is surrounded by features with similar values, either high or low.
Feature is part of a cluster.
Statistically significant clusters can consist of high values (HH) or low values (LL)
Negative I value:
Feature is surrounded by features with dissimilar values.
Feature is an outlier.
Statistically significant outliers can be a feature with a high value surrounded by features with low values (HL) or a feature with a low value surrounded by features with high values (LH).

26. Anselin Local Moran’s I
What do the dark blue and bright red splotches tell us in the map on the right?
Census tracts for percentage 65 and above
Z-scores
I values

27. Getis-Ord General G Global statistic
Indicates that high or low values are clustered
The value of the target feature itself is not included in the equation so it is useful to see the effect of the target feature on the surrounding area, such as for the dispersion of a disease.
Works best when either high or low values are clustered (but not both).
See ESRI online help for the equation.
Provides one statistic so if there are both high and low clusters then stat could be inaccurate.

28. Getis-Ord General G
High G score: Statistically significant clustering of high values.
Low G value: Slight clustering of low values.

29. Hot Spot Analysis (Getis-Ord Gi*)
Local version of the G statistic
The value of the target feature is included in analysis, which shows where hot spots (clusters of high values) or cold spots (clusters of low values) exist in the area.
To be statistically significant, the hot spot or cold spot will have a high/low value and be surrounded by other features with high/low values.
See ESRI online help for the equation.
Local equivalent of getis-ord general G

30. Hot Spot Analysis (Getis-Ord Gi*)
Gi* values
Z-scores
G=high value=hot spots
G=low value=cold spots

31. G statistics vs. Local Moran’s I
G statistics are useful when negative spatial autocorrelation (outliers) is negligible.
Local Moran’s I calculates spatial outliers.

32. G statistics in Geoda
Gi > The value itself at a location (i) is not included in the analysis
Gi* > The value (i) is included in the numerator and denominator.

33. Permutations in Geoda
Permutation inference is shuffling values around and re-computing statistics each time with a different set of random numbers to construct a reference distribution.
Permutations are used to determine how likely it would be to observe the Moran’s I value of an actual distribution under conditions of spatial randomness.
P-values are dependent on the number of permutations so they are “pseudo p-values”

34. Reference Distribution
Geoda generates a historgram of the Moran’s I values compared to the observed Moran’s I.

35. Bivariate Moran’s I An option in Geoda
It tests 2 variables: the correlation between a given variable (x) at a location and a different variable (y) at surrounding locations.
Results are difficult to interpret.
It is useful in examining the range of interaction provided x and y are correlated at the same location.

36. Resources ESRI Spatial Statistics Website:
Geoda Workbook:
ESRI Spatial Statistics Tool help:

37. ArcMap Help Links Spatial Autocorrelation (Moran’s I)
Anselin Local Moran’s I
Getis Ord General G
Hot Spot Analysis (Getis Ord Gi*)