Jennie Murack
murack@mit.edu
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
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.
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?
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
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
Types of data often analyzed
 Location of crimes, animals, retail, industry, etc.
 Land cover
 Land use
 Census/social data
Software
 ArcGIS
 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:
http://geodacenter.asu.edu/
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.
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
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)
Types of Contiguity
 Rook = Share edges
 Bishop = share corners
 Queen = share edges or corners
 Secondary order contiguity = neighbor of neighbor
Image from: http://www.lpc.uottawa.ca/publications/moransi/moran.htm
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)
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
Ripley’s K-function
Assaults are clustered until about 13,000 ft.
and then dispersed beyond 15,000 ft.
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)
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
Spatial Autocorrelation (Moran’s I)
Calculates I values:
I=0=random distribution
I<0=Values dispersed
0<I=Values clustered
Clustered
Dispersed
Random
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
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.
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).
Anselin Local Moran’s I
Census tracts
for percentage
65 and above
I values
Z-scores
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.
Getis-Ord General G
High G score:
Statistically significant
clustering of high
values.
Low G value: Slight
clustering of low
values.
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.
Hot Spot Analysis (Getis-Ord Gi*)
Gi* values
Z-scores
•G=high value=hot spots
•G=low value=cold spots
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.
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.
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”
Reference Distribution
 Geoda generates a historgram of the Moran’s I values
compared to the observed Moran’s I.
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.
Resources
 ESRI Spatial Statistics Website:
http://blogs.esri.com/Dev/blogs/geoprocessing/archiv
e/2010/07/13/Spatial-Statistics-Resources.aspx
 Geoda Workbook:
https://geodacenter.asu.edu/system/files/geodaworkb
ook.pdf
 ESRI Spatial Statistics Tool help:
http://resources.arcgis.com/en/help/main/10.1/index.
html#/An_overview_of_the_Spatial_Statistics_toolbox
/005p00000002000000/
ArcMap Help Links
 Spatial Autocorrelation (Moran’s I)
 http://resources.arcgis.com/en/help/main/10.1/index.html
 Anselin Local Moran’s I
 http://resources.arcgis.com/en/help/main/10.1/index.html
 Getis Ord General G
 http://resources.arcgis.com/en/help/main/10.1/index.html
 Hot Spot Analysis (Getis Ord Gi*)
 http://resources.arcgis.com/en/help/main/10.1/index.html