Local Spatial StatisticsLocal statistics are developed to measure dependence in only a portion ofthe 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 stationarity3. 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 AutocorrelationType 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

ExpectedKObservedKConfidence Env.

K FunctionClustered

Dispersed

Distance1000900800700600500400300200100

L(d)

1100

1000

900

800

700

600

500

400

300

200

100

ExpectedKObservedKConfidence Env.

K FunctionClustered

Dispersed

Distance1000900800700600500400300200100

L(d)

1200

1100

1000

900

800

700

600

500

400

300

200

100

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 MapFraction Hispanic Polygon ContiguityI = 0.83, Z = 19.3

High/Low Clustering

Quantile MapAverage Family SizePolygon Contiguity

I = 0.6; Z = 14.1

Anselin Local Moran Ii Cluster and Outlier Analysis

• Developed by Anselin (1995)

jiji

n

ijjj

i

n

jjiii

nwIE

Xn

Xx

s

ijXxsXxI

)1/()(

)1(

)(

),)/)((

2

2

,12

1

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 MapFraction Hispanic Polygon ContiguityI = 0.83, Z = 19.3

Anselin Local Moran Ii Cluster and Outlier Analysis

Quantile MapMed_AgePolygon ContiguityI = 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 MapFraction Hispanic Polygon ContiguityI = 0.83, Z = 19.3

Getis-Ord G Statistic

Quantile MapMed_AgePolygon ContiguityI = 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 MoranHR90 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.