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Local Spatial Statistics

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’

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Local Spatial Statistics

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  1. 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)

  2. 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-OrdGi*)

  3. Taxonomy of Autocorrelation

  4. Weighted Ripley K • Weighted Points • Evaluates Pattern of the Weighted Values • Must Use Confidence Intervals

  5. High/Low Clustering

  6. High/Low Clustering • To determine weights use: • Select Fixed Distance • Polygon Contiguity • K Nearest Neighbors • Delauny Triangulation • Select None for the Standardization parameter.

  7. High/Low Clustering • Quantile Map • Fraction Hispanic • Polygon Contiguity • I = 0.83, Z = 19.3

  8. High/Low Clustering • Quantile Map • Average Family Size • Polygon Contiguity • I = 0.6; Z = 14.1

  9. Anselin Local Moran Ii Cluster and Outlier Analysis • Developed by Anselin (1995)

  10. 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).

  11. Anselin Local Moran Ii Cluster and Outlier Analysis • Quantile Map • Fraction Hispanic • Polygon Contiguity • I = 0.83, Z = 19.3

  12. Anselin Local Moran Ii Cluster and Outlier Analysis • Quantile Map • Med_Age • Polygon Contiguity • I = 0.48, Z = 11.3

  13. 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.

  14. Gi* Statistic

  15. 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.

  16. Getis-Ord G Statistic • Quantile Map • Fraction Hispanic • Polygon Contiguity • I = 0.83, Z = 19.3

  17. Getis-Ord G Statistic • Quantile Map • Med_Age • Polygon Contiguity • I = 0.48, Z = 11.3

  18. Getis-Ord G Statistic vs Local Moran I

  19. 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”.

  20. Homicide rate per 100,000 (1990)

  21. Log Transformation (1 + HR90)

  22. Z(I) = 42.45

  23. Local Indicators of Spatial Association

  24. Bivariate MoranHR90 vs. Gini index of family income inequality

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

  26. 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.

  27. Moran LISA Scatter PlotsNumber of P. velutina plants within 5 X 5-m quadrats

  28. 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.

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