Local Indicators of Spatial Autocorrelation (LISA)

Autocorrelation Distance

Conceptualization of Spatial Relationships

• INVERSE_DISTANCE—Nearby neighboring features have a larger influence on the computations for a target feature than features that are far away.

• INVERSE_DISTANCE_SQUARED—Same as INVERSE_DISTANCE except that the slope is sharper, so influence drops off more quickly.

• FIXED_DISTANCE_BAND—Each feature is analyzed within the context of neighboring features. Neighboring features inside the specified critical distance receive a weight of 1.

• ZONE_OF_INDIFFERENCE—Features within the specified critical distance of a target feature receive a weight of 1 and once the critical distance is exceeded, weights diminish with distance.

• CONTIGUITY_EDGES_ONLY—Only neighboring polygon features that share a boundary or overlap will influence computations for the target polygon feature.

• CONTIGUITY_EDGES_CORNERS—Polygon features that share a boundary, share a node, or overlap will influence computations for the target polygon feature.

• GET_SPATIAL_WEIGHTS_FROM_FILE—Spatial relationships are defined in a spatial weights file. The path to the spatial weights file is specified in the Weights Matrix File

Distance Tools

• Calculate Distance Band from Neighbor Count• Returns the minimum, the maximum, and the average distance to the

specified Nth nearest neighbor (N is an input parameter) for a set of features.

• Incremental Spatial Autocorrelation• Measures spatial autocorrelation for a series of distances and optionally

creates a line graph of those distances and their corresponding z-scores.• Z-scores reflect the intensity of spatial clustering, and statistically significant

peak z-scores indicate distances where spatial processes promoting clustering are most pronounced. • These peak distances are often appropriate values to use for tools with a

Distance Band or Distance Radius parameter.

Random Pottery Example• Calculate Distance Band from Neighbor Count• Minimum Distance – 200 meters• Average Distance – 561 meters

• Can be used as an interval to determine correlation distances • Maximum Distance – 1166 meters

• The start to determine correlation distances

• Incremental Spatial Autocorrelation • Beginning Distance: 1200• Distance Increment: 600• Euclidean Distance

Random Pottery Example• Incremental Spatial Autocorrelation

OID Field1 Distance MoransI ExpectedI Variance z_score

0 0 1200 0.144238 -0.010753 0.003715 2.542825

1 0 1800 0.162113 -0.010753 0.001597 4.325894

2 0 2400 0.156748 -0.010753 0.000892 5.609849

3 0 3000 0.146881 -0.010753 0.000516 6.941486

4 0 3600 0.135467 -0.010753 0.000337 7.96461

5 0 4200 0.101642 -0.010753 0.000216 7.651647

6 0 4800 0.069922 -0.010753 0.000141 6.784276

7 0 5400 0.055625 -0.010753 0.000091 6.939656

8 0 6000 0.037376 -0.010753 0.000062 6.119467

9 0 6600 0.022392 -0.010753 0.000038 5.363339

Hot Spot Analysis (Getis-Ord Gi*)

Tucson Census Tract Data – Fraction of Hispanic

• Calculate Distance Band from Neighbor Count• Minimum Distance: 626 meters• Average Distance: 1630 meters• Maximum Distance: 6814 meters

• Incremental Spatial AutocorrelationOID Field1 Distance MoransI ExpectedI Variance z_score

0 0 7000 0.651477 -0.005988 0.000303 37.784478

1 0 8700 0.571928 -0.005988 0.000188 42.146948

2 0 10400 0.488893 -0.005988 0.000122 44.834375

3 0 12100 0.402285 -0.005988 0.000082 45.16891

4 0 13800 0.326296 -0.005988 0.000055 44.765833

5 0 15500 0.248237 -0.005988 0.000038 41.297881

6 0 17200 0.180755 -0.005988 0.000026 36.455985

7 0 18900 0.125267 -0.005988 0.000019 30.38139

8 0 20600 0.083063 -0.005988 0.000013 24.255486

9 0 22300 0.050116 -0.005988 0.00001 17.979835

High-Low ClusteringFixed Distance Extent Contiguity Edges Corners

Hot Spot Analysis

Fixed Distance Extent Contiguity Edges Corners