Spatial Autocorrelation (Global Moran's I) (Spatial Statistics)
Summary
Measures spatial autocorrelation based on feature locations and attribute values using the Global Moran's I statistic.
You can access the results of this tool (including the optional report file) from the Results window. If you disable background processing, results will also be written to the Progress dialog box.
Learn more about how Spatial Autocorrelation (Global Moran's I) works
Illustration
Usage

The Spatial Autocorrelation tool returns five values: the Moran's I Index, Expected Index, Variance, zscore, and pvalue. These values are accessible from the Results window and are also passed as derived output values for potential use in models or scripts. Optionally, this tool will create an HTML file with a graphical summary of results. Doubleclicking on the HTML file in the Results window will open the HTML file in the default Internet browser. Rightclicking on the Messages entry in the Results window and selecting View will display the results in a Message dialog box. If you execute this tool in the foreground, output values will also be displayed in the progress dialog box.
Note: If this tool is part of a custom model tool, the HTML link will only appear in the Results window if it is set as a model parameter prior to running the tool.
 For best display of HTML graphics, ensure your monitor is set to 96 DPI.

Given a set of features and an associated attribute, the Spatial Autocorrelation tool evaluates whether the pattern expressed is clustered, dispersed, or random. When the zscore or pvalue indicates statistical significance, a positive Moran's I index value indicates tendency toward clustering while a negative Moran's I index value indicates tendency toward dispersion.

The Global Moran's I tool calculates a zscore and pvalue to indicate whether or not you can reject the null hypothesis. In this case, the null hypothesis states that feature values are randomly distributed across the study area.

The zscore is based on the randomization null hypothesis computation. For more information on zscores, see What is a zscore? What is a pvalue?

The Input Field should contain a variety of values. The math for this statistic requires some variation in the variable being analyzed; it cannot solve if all input values are 1, for example. If you want to use this tool to analyze the spatial pattern of incident data, consider aggregating your incident data.

Calculations based on either Euclidean or Manhattan distance require projected data to accurately measure distances.
This tool will optionally create an HTML file summarizing results. HTML files will not automatically appear in the Catalog window. If you want HTML files to be displayed in Catalog, open the ArcCatalog application, select the Customize menu option, click ArcCatalog Options, and select the File Types tab. Click on the New Type button and specify HTML for File Extension.

For line and polygon features, feature centroids are used in distance computations. For multipoints, polylines, or polygons with multiple parts, the centroid is computed using the weighted mean center of all feature parts. The weighting for point features is 1, for line features is length, and for polygon features is area.

Your choice for the Conceptualization of Spatial Relationships parameter should reflect inherent relationships among the features you are analyzing. The more realistically you can model how features interact with each other in space, the more accurate your results will be. Recommendations are outlined in Selecting a Conceptualization of Spatial Relationships: Best Practices. Here are some additional tips:
 FIXED_DISTANCE_BAND
The default Distance Band or Threshold Distance will ensure each feature has at least one neighbor, and this is important. But often, this default will not be the most appropriate distance to use for your analysis. Additional strategies for selecting an appropriate scale (distance band) for your analysis are outlined in Selecting a fixed distance band value.
 INVERSE_DISTANCE or INVERSE_DISTANCE_SQUARED
When zero is entered for the Distance Band or Threshold Distance parameter, all features are considered neighbors of all other features; when this parameter is left blank, the default distance will be applied.
Weights for distances less than 1 become unstable when they are inverted. Consequently, the weighting for features separated by less than 1 unit of distance (common with Geographic Coordinate System projections) are given a weight of 1.
Caution:Analysis on features with a Geographic Coordinate System projection is not recommended when you select any of the inverse distancebased spatial conceptualization methods (INVERSE_DISTANCE, INVERSE_DISTANCE_SQUARED, or ZONE_OF_INDIFFERENCE).
For the inverse distance options (INVERSE_DISTANCE, INVERSE_DISTANCE_SQUARED, or ZONE_OF_INDIFFERENCE), any two points that are coincident will be given a weight of one to avoid zero division. This assures features are not excluded from analysis.
 FIXED_DISTANCE_BAND

Additional options for the Conceptualization of Spatial Relationships parameter, including spacetime relationships, are available using the Generate Spatial Weights Matrix or Generate Network Spatial Weights tools. To take advantage of these additional options, use one of these tools to construct the spatial weights matrix file prior to analysis; select GET_SPATIAL_WEIGHTS_FROM_FILE for the Conceptualization of Spatial Relationships parameter; and for the Weights Matrix File parameter, specify the path to the spatial weights file you created.

Map layers can be used to define the Input Feature Class. When using a layer with a selection, only the selected features are included in the analysis.
 Note:
 If this tool is part of a custom model tool, the HTML link will only appear in the Results window if it is set as a model parameter prior to running the tool.
 For best display of HTML graphics, ensure your monitor is set to 96 DPI.
If you provide a Weights Matrix File with an SWM extension, this tool is expecting a spatial weights matrix file created using either the Generate Spatial Weights Matrix or Generate Network Spatial Weights tools; otherwise, this tool is expecting an ASCII formatted spatial weights matrix file. In some cases, behavior is different depending on which type of spatial weights matrix file you use:
 ASCIIformatted spatial weights matrix files:
 Weights are used as is. Missing featuretofeature relationships are treated as zeros.
 If the weights are row standardized, results will likely be incorrect for analyses on selection sets. If you need to run your analysis on a selection set, convert the ASCII spatial weights file to an SWM file by reading the ASCII data into a table, then using the CONVERT_TABLE option with the Generate Spatial Weights Matrix tool.
 SWMformatted spatial weights matrix file:
 If the weights are row standardized, they will be restandardized for selection sets; otherwise, weights are used as is.
Running your analysis with an ASCII formatted spatial weights matrix file is memory intensive. For analyses on more than 5,000 features, consider converting your ASCIIformatted spatial weights matrix file into an SWM formatted file. First put your ASCII weights into a formatted table (using Excel, for example). Next, run the Generate Spatial Weights Matrix tool using CONVERT_TABLE for the Conceptualization of Spatial Relationships parameter. The output will be an SWMformatted spatial weights matrix file.
For polygon features, you will almost always want to choose Row for the Row Standardization parameter. Row Standardization mitigates bias when the number of neighbors each feature has is a function of the aggregation scheme or sampling process, rather than reflecting the actual spatial distribution of the variable you are analyzing.

The Modeling Spatial Relationships help topic provides additional information about this tool's parameters.
In ArcGIS 10, optional graphical output is no longer displayed automatically. Instead, an HTML file summarizing results is created. To view results, doubleclick on the HTML file in the Results window. Custom scripts or model tools created prior to ArcGIS 10 that use this tool may need to be rebuilt. To rebuild these custom tools, open them, remove the Display Results Graphically parameter, then resave.
It is possible to run out of memory when you run this tool. This generally occurs when you select Conceptualization of Spatial Relationships and/or Distance Band or Threshold Distance resulting in features having many, many neighbors. You generally do not want to define spatial relationships so that features have thousands of neighbors. You want all features to have at least one neighbor and almost all features to have at least eight neighbors.
When using shapefiles, keep in mind that they cannot store null values. Tools or other procedures that create shapefiles from nonshapefile inputs may store or interpret null values as zero. In some cases, nulls are stored as very large negative values in shapefiles. This can lead to unexpected results. See Geoprocessing considerations for shapefile output for more information.
In ArcGIS version 9.2, the Global standardization option was removed. Global standardization returns the same results as no standardization. Models built with previous versions of ArcGIS that use the Global standardization option may need to be rebuilt.
Syntax
Parameter  Explanation  Data Type 
Input_Feature_Class 
The feature class for which spatial autocorrelation will be calculated.  Feature Layer 
Input_Field 
The numeric field used in assessing spatial autocorrelation.  Field 
Generate_Report (Optional) 
 Boolean 
Conceptualization_of_Spatial_Relationships 
Specifies how spatial relationships among features are conceptualized.
 String 
Distance_Method 
Specifies how distances are calculated from each feature to neighboring features.
 String 
Standardization 
Row standardization is recommended whenever the distribution of your features is potentially biased due to sampling design or an imposed aggregation scheme.
 String 
Distance_Band_or_Threshold_Distance (Optional) 
Specifies a cutoff distance for Inverse Distance and Fixed Distance options. Features outside the specified cutoff for a target feature are ignored in analyses for that feature. However, for Zone of Indifference, the influence of features outside the given distance is reduced with distance, while those inside the distance threshold are equally considered. The distance value entered should match that of the output coordinate system. For the Inverse Distance conceptualizations of spatial relationships, a value of 0 indicates that no threshold distance is applied; when this parameter is left blank, a default threshold value is computed and applied. This default value is the Euclidean distance that ensures every feature has at least one neighbor. This parameter has no effect when Polygon Contiguity or Get Spatial Weights From File spatial conceptualizations are selected.  Double 
Weights_Matrix_File (Optional) 
The path to a file containing weights that define spatial, and potentially temporal, relationships among features.  File 
Code Sample
The following Python window script demonstrates how to use the SpatialAutocorrelation tool.
import arcpy
arcpy.env.workspace = r"c:\data"
arcpy.SpatialAutocorrelation_stats("olsResults.shp", "Residual","NO_REPORT",
"GET_SPATIAL_WEIGHTS_FROM_FILE","EUCLIDEAN DISTANCE",
"NONE", "#","euclidean6Neighs.swm")
The following standalone Python script demonstrates how to use the SpatialAutocorrelation tool.
# Analyze the growth of regional per capita incomes in US
# Counties from 1969  2002 using Ordinary Least Squares Regression
# Import system modules
import arcpy
# Set the geoprocessor object property to overwrite existing outputs
arcpy.gp.overwriteOutput = True
# Local variables...
workspace = r"C:\Data"
try:
# Set the current workspace (to avoid having to specify the full path to the feature classes each time)
arcpy.workspace = workspace
# Growth as a function of {log of starting income, dummy for South
# counties, interaction term for South counties, population density}
# Process: Ordinary Least Squares...
ols = arcpy.OrdinaryLeastSquares_stats("USCounties.shp", "MYID",
"olsResults.shp", "GROWTH",
"LOGPCR69;SOUTH;LPCR_SOUTH;PopDen69",
"olsCoefTab.dbf",
"olsDiagTab.dbf")
# Create Spatial Weights Matrix (Can be based off input or output FC)
# Process: Generate Spatial Weights Matrix...
swm = arcpy.GenerateSpatialWeightsMatrix_stats("USCounties.shp", "MYID",
"euclidean6Neighs.swm",
"K_NEAREST_NEIGHBORS",
"#", "#", "#", 6)
# Calculate Moran's I Index of Spatial Autocorrelation for
# OLS Residuals using a SWM File.
# Process: Spatial Autocorrelation (Morans I)...
moransI = arcpy.SpatialAutocorrelation_stats("olsResults.shp", "Residual",
"NO_REPORT", "GET_SPATIAL_WEIGHTS_FROM_FILE",
"EUCLIDEAN_DISTANCE", "NONE", "#",
"euclidean6Neighs.swm")
except:
# If an error occurred when running the tool, print out the error message.
print arcpy.GetMessages()
Environments
 Output Coordinate System
Feature geometry is projected to the Output Coordinate System prior to analysis. All mathematical computations are based on the Output Coordinate System spatial reference.