Spatial Statistics I
RESM 575
Spring 2011
Lecture 10
Spatial Statistics
Measuring geographic
distributions
Testing statistical significance
Identifying patterns
Measuring geographic distributions
Identify spatial characteristics of a distribution
 Where is the center?
 What feature is most central?
 How are features dispersed around the
center?
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ArcTools
in
ArcGIS
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Where is the center?

Mean Center
tool
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Computes
the average
X and Y
coordinates
of all
features
Creates a
new point
feature
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Example
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Example
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Mean center tool

More common use:

To compare distributions of different types of
features or to find the center of features based on
an attribute value
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Example
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Example
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Example
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What is the most central feature?

Central feature tool

Identifies the most centrally located feature

Feature having the lowest total distance to all other
features
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Example
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Example
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What is the most central feature?


More interestingly is by adding a weight in the
analysis such as population
We are now finding not just which site is most
central but which is the most accessible to
the greatest number of people.
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Example
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Measuring feature distribution

Standard distance tool

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Measures distribution of features around the
mean
Result is a summary statistic representing
distance
If circle is large, incidents are widespread
If small, incidents are more localized
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Example
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Distributional trends

Directional distribution (standard ellipse) tool

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Identify spatial trends in the distribution of
features
Uses

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
Compare distributions
Examine different time periods
Show compactness and orientation
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Example
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Testing statistical significance

The next section of identifying patterns or
later spatial relationships allows us to perform
significance tests on the results before
accepting them
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Using significance tests with spatial data


Spatial data contradicts some of the
assumptions of inferential statistics
You need to be aware of these limitations!
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Assumptions

Testing a random sample



With GIS data in a database you may not know if
the data was randomly sampled
How large the sample is in relation to the
population?
Even with randomness assumed, spatial data
often violates the independence of
observations in a sample
Spatial data is rarely evenly distributed across a region
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For spatial pattern analysis…


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The null hypothesis is that features are
evenly distributed across the study area
Hard to imagine this being true
You have to make one of two common
sampling assumptions: randomization or
normalization
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Identifying Patterns
Why study patterns?
Range from completely clustered to completely
dispersed

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Identifying patterns (applications)
Forestry applications
USFS may measure the pattern of clear cuts to
ensure sufficient contiguous forest habitat remaining
Agency may allow a level of clustering of clear cuts
and then make sure it is not exceeded
Wildlife studies
if population is dispersed then species can live in a
wide range of habitats, if clustered then it has very
specific habitat requirements
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Goal for analyzing spatial patterns


Are there underlying spatial processes
influencing the locations of our features?
Are our features randomly located throughout
the study area, or are they displaying a
clustering or dispersed pattern?
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Approaches for analyzing spatial patterns

Approach #1 Global
calculations

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Identifies overall patterns or
trends in the data
Effective for complex messy
data
Interested in broad overall
results
Work by comparing feature
locations and/or attributes to
a theoretical random
distribution to determine if
you have statistically
significant clustering or
dispersion
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Analyzing spatial patterns


Can be found with the Average Nearest Neighbor
which does not require specifying an attribute, or
If based on an attribute we can test for Spatial
autocorrelation using the (Moran’s I) tool

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Things that are closer are more alike than things that are
not
Measures similarity of neighboring features
Identifies if features are clustered or dispersed
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After tool runs, go to Geoprocessing - > Results
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Double click to open the html
report for your latest run
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After tool runs, go to Geoprocessing - > Results
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Global SA interpretation
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I > 0 positive spatial autocorrelation
I < 0 negative spatial autocorrelation
The more positive or more negative, the greater
amount of spatial autocorrelation
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2nd Approach
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Local calculations
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Identify the extent and
locations of clustering
Answer where do we
have spatial clustering
Process every feature
within the context of
its neighboring
features in order to
determine whether it
represents a spatial
outlier, or if part of a
statistically significant
spatial cluster
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Locate the hot spots

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This is a local question that requires a hot spot
analysis (Getis-Ord Gi*) tool
Indicates the extent to which each feature is
surrounded by similarly high or low values
Where do features with similar attribute values
cluster spatially together
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Getis-Ord Gi* tool

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Identifies where clustering occurs in both high
and low values
Calculates a Z score for each feature

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High Z = hot spot (when a feature has a high
value and it is surrounded by other features with
high values)
Low Z = cold spot (when we have features with
low values surrounded by other features with low
values)
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Notes on the Z score

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Z score is a measure of standard deviation
It is a reference value that’s associated with a
standard normal distribution
A very high or low Z score would be found in
the tails
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More notes on Z score
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A very high or low Z score means that the
pattern deviates significantly from a
hypothetical random pattern
For example, when using a 95% CI, Z scores
are -1.96 and +1.96
If Z is between these -1.96 and +1.96 you
can’t reject the null
You are seeing one version of a random
pattern
If very high or low (ie -2.5 or +5.4) you have a
pattern that’s too unusual to be a pattern of
random choice so we reject the null hypoth
REMEMBER: The null hypothesis is that features are evenly
distributed across the study area
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Reference


Mitchell, A. 2005. ESRI Guide to GIS
Analysis, Volume 2. ESRI press, Redlands,
CA.
ArcGIS 10 Help
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