Cluster and Outlier Analysis (Anselin Local Moran's I) (Spatial Statistics)

Summary

Given a set of weighted features, identifies statistically significant hot spots, cold spots, and spatial outliers using the Anselin Local Moran's I statistic.

Learn more about how Cluster and Outlier Analysis (Anselin Local Moran's I) works

Illustration

Usage

• This tool creates a new Output Feature Class with the following attributes for each feature in the Input Feature Class: Local Moran's I index, z-score, pseudo p-value, and cluster/outlier type (COType).

• The z-scores and p-values are measures of statistical significance which tell you whether or not to reject the null hypothesis, feature by feature. In effect, they indicate whether the apparent similarity (a spatial clustering of either high or low values) or dissimilarity (a spatial outlier) is more pronounced than one would expect in a random distribution. The p-values and z-scores in the Output Feature Class do not reflect any FDR (False Discovery Rate) corrections.

• A high positive z-score for a feature indicates that the surrounding features have similar values (either high values or low values). The COType field in the Output Feature Class will be HH for a statistically significant cluster of high values and LL for a statistically significant cluster of low values.

• A low negative z-score (for example, less than -3.96) for a feature indicates a statistically significant spatial data outlier. The COType field in the Output Feature Class will indicate if the feature has a high value and is surrounded by features with low values (HL) or if the feature has a low value and is surrounded by features with high values (LH).

• The COType field will always indicate statistically significant clusters and outliers for a 95 percent confidence level. Only statistically significant features have values for the COType field. When you check the optional Apply False Discovery Rate (FDR) Correction parameter, statistical significance is based on a corrected 95 percent confidence level.

• Default rendering for the Output Feature Class is based on the values in the COType field.

• The output of this tool also includes a histogram charting the value of the Input Field and a Moran's scatterplot. These charts can be accessed from the Contents pane under the Output Feature Class.

• Permutations are used to determine how likely it would be to find the actual spatial distribution of the values you are analyzing. For each permutation, the neighborhood values around each feature are randomly rearranged and the Local Moran's I value calculated. The result is a reference distribution of values that is then compared to the actual observed Moran's I to determine the probability that the observed value could be found in the random distribution. The default is 499 permutations; however, the random sample distribution is improved with increasing permutations, which improves the precision of the pseudo p-value.

• If the Number of Permutations parameter is set to 0, the result is a traditional p-value instead of a pseudo p-value and the z-score is based on the randomization null hypothesis computation. For more information on z-scores and p-values, see What is a z-score? What is a p-value?

• When the Input Feature Class is not projected (that is, when coordinates are given in degrees, minutes, and seconds) or when the output coordinate system is set to a Geographic Coordinate System, distances are computed using chordal measurements. Chordal distance measurements are used because they can be computed quickly and provide very good estimates of true geodesic distances, at least for points within about 30 degrees of each other. Chordal distances are based on an oblate spheroid. Given any two points on the earth's surface, the chordal distance between them is the length of a line, passing through the three-dimensional earth, to connect those two points. Chordal distances are reported in meters.

  Caution:

  Be sure to project your data if your study area extends beyond 30 degrees. Chordal distances are not a good estimate of geodesic distances beyond 30 degrees.

• When chordal distances are used in the analysis, the Distance Band or Threshold Distance parameter, if specified, should be given in meters.

• 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.

• 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. The Optimized Hot Spot Analysis tool may also be used to analyze the spatial pattern of incident data.

  Note:

  Incident data are points representing events (crime, traffic accidents) or objects (trees, stores) where your focus is on presence or absence rather than some measured attribute associated with each point.

• 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

    Uses a Distance Band or Threshold Distance and it will ensure each feature has at least one neighbor. 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 are given a weight of 1.

    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 1 to avoid zero division. This assures features are not excluded from analysis.

• Additional options for the Conceptualization of Spatial Relationships parameter, including space-time relationships, are available using the Generate Spatial Weights Matrix tool. To take advantage of these additional options, construct a 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.

• More information about space-time cluster analysis is provided in the Space-Time Analysis documentation.

• 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.

• If you provide a Weights Matrix File with a .swm extension, this tool is expecting a spatial weights matrix file created using the Generate Spatial Weights Matrix tool; 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:

  • ASCII-formatted spatial weights matrix files:
    • Weights are used as is. Missing feature-to-feature 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.
  • SWM-formatted 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 ASCII-formatted 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 SWM-formatted spatial weights matrix file.

• The Output Feature Class is automatically added to the table of contents with default rendering applied to the COType field. The rendering applied is defined by a layer file in <ArcGIS Pro>\Resources\ArcToolBox\Templates\Layers. You can reapply the default rendering, if needed, by using the Apply Symbology From Layer tool.

• The Output Feature Class includes a SOURCE_ID field which allows you to Join it to the Input Feature Class, if needed.

• The Modeling Spatial Relationships help topic provides additional information about this tool's parameters.

Caution:

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.

• When using this tool in Python scripts, the result object returned from tool execution has the following outputs:

  Position	Description	Data Type
  0	Output feature class	Feature Class
  1	Index field name	Field
  2	ZScore field name	Field
  3	Probability field name	Field
  4	COType field name	Field
  5	Source ID field name	Field

Parameters

Dialog

Label	Explanation	Data Type
Input Feature Class	The feature class for which cluster and outlier analysis will be performed.	Feature Layer
Input Field	The numeric field to be evaluated.	Field
Output Feature Class	The output feature class to receive the results fields.	Feature Class
Conceptualization of Spatial Relationships	Specifies how spatial relationships among features are defined. 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, and only a target feature's closest neighbors will exert substantial influence on computations for that feature. Fixed distance band—Each feature is analyzed within the context of neighboring features. Neighboring features inside the specified critical distance (Distance Band or Threshold Distance) receive a weight of one and exert influence on computations for the target feature. Neighboring features outside the critical distance receive a weight of zero and have no influence on a target feature's computations. Zone of indifference—Features within the specified critical distance (Distance Band or Threshold Distance) of a target feature receive a weight of one and influence computations for that feature. Once the critical distance is exceeded, weights (and the influence a neighboring feature has on target feature computations) diminish with distance. K nearest neighbors—The closest k features are included in the analysis. The number of neighbors (k) is specified by the Number of Neighbors parameter. 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 by a specified spatial weights file. The path to the spatial weights file is specified by the Weights Matrix File parameter.	String
Distance Method	Specifies how distances are calculated from each feature to neighboring features. Euclidean—The straight-line distance between two points (as the crow flies) Manhattan—The distance between two points measured along axes at right angles (city block); calculated by summing the (absolute) difference between the x- and y-coordinates	String
Standardization	Row standardization is recommended whenever the distribution of your features is potentially biased due to sampling design or an imposed aggregation scheme. None—No standardization of spatial weights is applied. Row—Spatial weights are standardized; each weight is divided by its row sum (the sum of the weights of all neighboring features).	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
Apply False Discovery Rate (FDR) Correction (Optional)	Specifies whether statistical significance will be assessed with or without FDR correction. Checked—Statistical significance will be based on the False Discovery Rate correction for a 95 percent confidence level. Unchecked—Features with p-values less than 0.05 will appear in the COType field reflecting statistically significant clusters or outliers at a 95 percent confidence level. This is the default.	Boolean
Number of Permutations (Optional)	The number of random permutations for the calculation of pseudo p-values. The default number of permutations is 499. If you choose 0 permutations, the standard p-value is calculated. 0—Permutations are not used and a standard p-value is calculated. 99—With 99 permutations, the smallest possible pseudo p-value is 0.01 and all other pseudo p-values will be multiples of this value. 199—With 199 permutations, the smallest possible pseudo p-value is 0.005 and all other possible pseudo p-values will be multiples of this value. 499—With 499 permutations, the smallest possible pseudo p-value is 0.002 and all other pseudo p-values will be multiples of this value. 999—With 999 permutations, the smallest possible pseudo p-value is 0.001 and all other pseudo p-values will be multiples of this value. 9999—With 9999 permutations, the smallest possible pseudo p-value is 0.0001 and all other pseudo p-values will be multiples of this value.	Long
Number of Neighbors (Optional)	The number of neighbors to include in the analysis.	Long

Derived Output

Label	Explanation	Data Type
Index Field Name	The index field name.	Field
ZScore Field Name	The z-score field name.	Field
Probability Field	The probability field name.	Field
Cluster Outlier Type	The cluster/outlier field name.	Field
Source ID	The source ID field name.	Field

Python

arcpy.stats.ClustersOutliers(Input_Feature_Class, Input_Field, Output_Feature_Class, Conceptualization_of_Spatial_Relationships, Distance_Method, Standardization, {Distance_Band_or_Threshold_Distance}, {Weights_Matrix_File}, {Apply_False_Discovery_Rate__FDR__Correction}, {Number_of_Permutations}, {number_of_neighbors})

Name	Explanation	Data Type
Input_Feature_Class	The feature class for which cluster and outlier analysis will be performed.	Feature Layer
Input_Field	The numeric field to be evaluated.	Field
Output_Feature_Class	The output feature class to receive the results fields.	Feature Class
Conceptualization_of_Spatial_Relationships	Specifies how spatial relationships among features are defined. 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, and only a target feature's closest neighbors will exert substantial influence on computations for that feature. FIXED_DISTANCE_BAND—Each feature is analyzed within the context of neighboring features. Neighboring features inside the specified critical distance (Distance_Band_or_Threshold_Distance) receive a weight of one and exert influence on computations for the target feature. Neighboring features outside the critical distance receive a weight of zero and have no influence on a target feature's computations. ZONE_OF_INDIFFERENCE—Features within the specified critical distance (Distance_Band_or_Threshold_Distance) of a target feature receive a weight of one and influence computations for that feature. Once the critical distance is exceeded, weights (and the influence a neighboring feature has on target feature computations) diminish with distance. K_NEAREST_NEIGHBORS—The closest k features are included in the analysis. The number of neighbors (k) is specified by the number_of_neighbors parameter. 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 by a specified spatial weights file. The path to the spatial weights file is specified by the Weights_Matrix_File parameter.	String
Distance_Method	Specifies how distances are calculated from each feature to neighboring features. EUCLIDEAN_DISTANCE—The straight-line distance between two points (as the crow flies) MANHATTAN_DISTANCE—The distance between two points measured along axes at right angles (city block); calculated by summing the (absolute) difference between the x- and y-coordinates	String
Standardization	Row standardization is recommended whenever the distribution of your features is potentially biased due to sampling design or an imposed aggregation scheme. NONE—No standardization of spatial weights is applied. ROW—Spatial weights are standardized; each weight is divided by its row sum (the sum of the weights of all neighboring features).	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
Apply_False_Discovery_Rate__FDR__Correction (Optional)	APPLY_FDR—Statistical significance will be based on the False Discovery Rate correction for a 95 percent confidence level. NO_FDR—Features with p-values less than 0.05 will appear in the COType field reflecting statistically significant clusters or outliers at a 95 percent confidence level (default).	Boolean
Number_of_Permutations (Optional)	The number of random permutations for the calculation of pseudo p-values. The default number of permutations is 499. If you choose 0 permutations, the standard p-value is calculated. 0—Permutations are not used and a standard p-value is calculated. 99—With 99 permutations, the smallest possible pseudo p-value is 0.01 and all other pseudo p-values will be multiples of this value. 199—With 199 permutations, the smallest possible pseudo p-value is 0.005 and all other possible pseudo p-values will be multiples of this value. 499—With 499 permutations, the smallest possible pseudo p-value is 0.002 and all other pseudo p-values will be multiples of this value. 999—With 999 permutations, the smallest possible pseudo p-value is 0.001 and all other pseudo p-values will be multiples of this value. 9999—With 9999 permutations, the smallest possible pseudo p-value is 0.0001 and all other pseudo p-values will be multiples of this value.	Long
number_of_neighbors (Optional)	The number of neighbors to include in the analysis.	Long

Derived Output

Name	Explanation	Data Type
Index_Field_Name	The index field name.	Field
ZScore_Field_Name	The z-score field name.	Field
Probability_Field	The probability field name.	Field
Cluster_Outlier_Type	The cluster/outlier field name.	Field
Source_ID	The source ID field name.	Field

Code sample

ClustersOutliers example 1 (Python window)

The following Python window script demonstrates how to use the ClustersOutliers tool.

import arcpy
arcpy.env.workspace = "c:/data/911calls"
arcpy.ClustersOutliers_stats("911Count.shp", "ICOUNT","911ClusterOutlier.shp",
                             "GET_SPATIAL_WEIGHTS_FROM_FILE","EUCLIDEAN_DISTANCE", 
                             "NONE","#", "euclidean6Neighs.swm","NO_FDR", 499)

ClustersOutliers example 2 (stand-alone script)

The following stand-alone Python script demonstrates how to use the ClustersOutliers tool.

# Analyze the spatial distribution of 911 calls in a metropolitan area
# using the Cluster-Outlier Analysis Tool (Anselin's Local Moran's I)

# Import system modules
import arcpy

# Set property to overwrite outputs if they already exist
arcpy.env.overwriteOutput = True

# Local variables...
workspace = r"C:\Data\911Calls"

try:
    # Set the current workspace 
    #  (to avoid having to specify the full path to the feature classes each time)
    arcpy.env.workspace = workspace

    # Copy the input feature class and integrate the points to snap
    # together at 500 feet
    # Process: Copy Features and Integrate
    cf = arcpy.CopyFeatures_management("911Calls.shp", "911Copied.shp")

    integrate = arcpy.Integrate_management("911Copied.shp #", "500 Feet")

    # Use Collect Events to count the number of calls at each location
    # Process: Collect Events
    ce = arcpy.CollectEvents_stats("911Copied.shp", "911Count.shp", "Count", "#")

    # Add a unique ID field to the count feature class
    # Process: Add Field and Calculate Field
    af = arcpy.AddField_management("911Count.shp", "MyID", "LONG", "#", "#", "#", "#",
                     														"NON_NULLABLE", "NON_REQUIRED", "#",
                     														"911Count.shp")
    
    cf = arcpy.CalculateField_management("911Count.shp", "MyID", "!FID!", "PYTHON")

    # Create Spatial Weights Matrix for Calculations
    # Process: Generate Spatial Weights Matrix... 
    swm = arcpy.GenerateSpatialWeightsMatrix_stats("911Count.shp", "MYID",
                        																											"euclidean6Neighs.swm",
                       																											 "K_NEAREST_NEIGHBORS",
                       															 												"#", "#", "#", 6) 

    # Cluster/Outlier Analysis of 911 Calls
    # Process: Local Moran's I
    clusters = arcpy.ClustersOutliers_stats("911Count.shp", "ICOUNT", 
                      																				  "911ClusterOutlier.shp", 
                        																				"GET_SPATIAL_WEIGHTS_FROM_FILE",
                        																				"EUCLIDEAN_DISTANCE", "NONE",
                       							 													"#", "euclidean6Neighs.swm", "NO_FDR", "499")

except arcpy.ExecuteError:
    # If an error occurred when running the tool, print out the error message.
    print(arcpy.GetMessages())

Environments

Current Workspace, Scratch Workspace, Output Coordinate System, Geographic Transformations, Qualified Field Names, Output has Z values, Default Output Z Value, Z Resolution, Z Tolerance, Output has M values, M Resolution, M Tolerance, XY Resolution, XY Tolerance, Random number generator

Output Coordinate System

Feature geometry is projected to the Output Coordinate System prior to analysis, so values entered for the Distance Band or Threshold Distance parameter should match those specified in the Output Coordinate System. All mathematical computations are based on the spatial reference of the Output Coordinate System. When the Output Coordinate System is based on degrees, minutes, and seconds, geodesic distances are estimated using chordal distances in meters.

Random number generator

The Random Generator Type used is always Mersenne Twister.

Licensing information

• Basic: Yes
• Standard: Yes
• Advanced: Yes