Local Spatial Statistics
Local statistics are developed to measure dependence in only a portion of
the area.
They measure the association between Xi and its neighbors up to a
specific distance from site i.
These statistics are well suited for:
1. Identify “hot spots’
2. Assess assumptions of stationarity
3. Identify distances beyond which no discernible association obtains.
Members of Local Indicator of Spatial Association (LISA)
Spatial Statistics Tools
•
•
•
•
High/Low Clustering (Getis-Ord General G)
Incremental Spatial Autocorrelation
Weighted Ripley K Function
Cluster and Outlier Analysis (Anselin Local
Morans I)
• Group Analysis
• Hot Spot Analysis (Getis-Ord Gi*)
Taxonomy of Autocorrelation
Type
Cross-Products
Differences Squared
Global,
Single Meas.
Moran
Geary
Global
Multiple Dist
Correlogram
Variogram
Local,
Multiple Dist
Gji, Gi*, Ii
Cji, K1ji, K2i
Weighted Ripley K
• Weighted Points
• Evaluates Pattern of the Weighted Values
• Must Use Confidence Intervals
K Function
K Function
Clustered
Clustered
ExpectedK
ExpectedK
ObservedK
ObservedK
Confidence Env.
Confidence Env.
1200
1100
1100
1000
1000
900
900
800
800
700
700
600
L(d)
L(d)
600
500
500
400
400
300
300
200
200
100
100
100
100
200
200
300
300
400
400
500
500
600
600
Distance
Distance
700
700
800
800
900
900
1000
1000
Dispersed
Dispersed
High/Low Clustering
High/Low Clustering
• To determine weights use:
–
–
–
–
Select Fixed Distance
Polygon Contiguity
K Nearest Neighbors
Delauny Triangulation
• Select None for the
Standardization parameter.
High/Low Clustering
Quantile Map
Fraction Hispanic
Polygon Contiguity
I = 0.83, Z = 19.3
High/Low Clustering
Quantile Map
Average Family Size
Polygon Contiguity
I = 0.6; Z = 14.1
Anselin Local Moran Ii
Cluster and Outlier Analysis
• Developed by Anselin (1995)
n
I i  ((xi  X ) / si2 ) x j  X ), j  i
j 1
n
s 
2
i
2
(
x

X
)
 j
j 1, j  i
(n  1)
X
E ( I i )   wij /(n  1)
j
2
Anselin Local Moran Ii
Cluster and Outlier Analysis
• Cluster Type (COType): distinguishes between a statistically
significant (0.05 level) cluster of high values (HH), cluster of
low values (LL), outlier in which a high value is surrounded
primarily by low values (HL), and outlier in which a low value
is surrounded primarily by high values (LH).
• Unique Feature - Local Moran Ii will identify statistically
significant spatial outliers (a high value surrounded by low
values or a low value surrounded by high values).
Anselin Local Moran Ii
Cluster and Outlier Analysis
Quantile Map
Fraction Hispanic
Polygon Contiguity
I = 0.83, Z = 19.3
Anselin Local Moran Ii
Cluster and Outlier Analysis
Quantile Map
Med_Age
Polygon Contiguity
I = 0.48, Z = 11.3
Getis-Ord G Statistic
• The null hypothesis is that the sum of values at
all the j sites within radius d of site i is not
more or less then expect by chance given all
the values in the entire study area.
• The Gi statistics does not include site i in
computing the sum.
• The Gi* statistic does include site i in
computing the sum.
Gi* Statistic
Getis-Ord G Statistic
• Interpretation
– The Gi* statistic returned for each feature in the
dataset is a z-score.
• For statistically significant positive z-scores, the larger
the z-score is, the more intense the clustering of high
values (hot spot).
• For statistically significant negative z-scores, the
smaller the z-score is, the more intense the clustering of
low values (cold spot).
– The Gi* statistic is a Z score.
Getis-Ord G Statistic
Quantile Map
Fraction Hispanic
Polygon Contiguity
I = 0.83, Z = 19.3
Getis-Ord G Statistic
Quantile Map
Med_Age
Polygon Contiguity
I = 0.48, Z = 11.3
Getis-Ord G Statistic vs Local Moran I
Problems
• Correlation Problem
– Overlapping samples of j, similar local statistics.
– Problem if statistical significance is sought.
• Small Sample Problem
– Statistics are based on a normal distribution, which is
unlikely for a small sample.
• Effects of Global Autocorrelation Problem
– If there is significant overall global autocorrelation the
local statistics will be less useful in detecting “hot spots”.
Homicide rate per 100,000 (1990)
Log Transformation (1 + HR90)
Z(I) = 42.45
Local Indicators of Spatial Association
Bivariate Moran
HR90 vs.
Gini index of family income inequality
Dawn Browning
• Disturbance, space, and time: Long-term mesquite (Prosopis velutina)
dynamics in Sonoran desert grasslands (1932 – 2006)
• Located on Santa Rita Experimental Range
Dawn Browning
• Trends in plant- and landscape-based aboveground P. velutina biomass
derived from field measurements of plant canopy area in 1932, 1948, and
2006.
Moran LISA Scatter Plots
Number of P. velutina plants
within 5 X 5-m quadrats
• Local indicator of spatial association (LISA) cluster maps and associated
Global Moran’s I values for P. velutina plant density within 5-m X 5-m
quadrats.