Local Spatial Statistics

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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
• Identify hot spots
• Assess assumptions of stationarity
• Identify distances beyond which no discernible
  association obtains.
• Members of Local Indicator of Spatial Association
  (LISA)

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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)

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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
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Weighted Ripley K

• Weighted Points
• Evaluates Pattern of the Weighted Values
• Must Use Confidence Intervals

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High/Low Clustering

•  

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High/Low Clustering

• To determine weights use
• Select Fixed Distance
• Polygon Contiguity
• K Nearest Neighbors
• Delauny Triangulation
• Select None for the Standardization parameter.

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High/Low Clustering
Quantile Map Fraction Hispanic Polygon
Contiguity I 0.83, Z 19.3
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High/Low Clustering
Quantile Map Average Family Size Polygon
Contiguity I 0.6 Z 14.1
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Anselin Local Moran Ii Cluster and Outlier
Analysis

• Developed by Anselin (1995)

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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).

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Anselin Local Moran Ii Cluster and Outlier
Analysis
Quantile Map Fraction Hispanic Polygon
Contiguity I 0.83, Z 19.3
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Anselin Local Moran Ii Cluster and Outlier
Analysis
Quantile Map Med_Age Polygon Contiguity I 0.48,
Z 11.3
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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.

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Gi Statistic
 
 
 
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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.

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Getis-Ord G Statistic
Quantile Map Fraction Hispanic Polygon
Contiguity I 0.83, Z 19.3
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Getis-Ord G Statistic
Quantile Map Med_Age Polygon Contiguity I 0.48,
Z 11.3
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Getis-Ord G Statistic vs Local Moran I
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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.

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Homicide rate per 100,000 (1990)
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Log Transformation (1 HR90)
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Z(I) 42.45
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Local Indicators of Spatial Association
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Bivariate MoranHR90 vs. Gini index of family
income inequality
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Dawn Browning

• Disturbance, space, and time Long-term mesquite
  (Prosopis velutina) dynamics in Sonoran desert
  grasslands (1932 2006)
• Located on Santa Rita Experimental Range

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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.

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Moran LISA Scatter PlotsNumber of P. velutina
plants within 5 X 5-m quadrats
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• Local indicator of spatial association (LISA)
  cluster maps and associated Global Morans I
  values for P. velutina plant density within 5-m X
  5-m quadrats.