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 TypeCross-ProductsDifferences - Squared Global, Single Meas. MoranGeary Global Multiple Dist CorrelogramVariogram Local, Multiple Dist G ji, G i *, I i C ji, K 1ji, K 2i

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

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 I i Cluster and Outlier Analysis Developed by Anselin (1995)

Anselin Local Moran I i 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 I i will identify statistically significant spatial outliers (a high value surrounded by low values or a low value surrounded by high values).

Anselin Local Moran I i Cluster and Outlier Analysis Quantile Map Fraction Hispanic Polygon Contiguity I = 0.83, Z = 19.3

Anselin Local Moran I i 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 G i statistics does not include site i in computing the sum. The G i * statistic does include site i in computing the sum.

G i * Statistic

Getis-Ord G Statistic Interpretation –The G i * 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 G i * 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.