Label | 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) | Specifies whether a graphical summary of result will be created as an .html file. - Checked—A graphical summary will be created.
- Unchecked—No graphical summary will be created. This is the default.
| Boolean |

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—This is the 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) to include in the analysis 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) will be used.
- Manhattan—The distance between two points measured along axes at right angles (city block) will be used. This is calculated by summing the (absolute) difference between the x- and y-coordinates
| String |

Standardization | Specifies whether standardization of spatial weights will be applied. 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). This is the default.
| String |

Distance Band or Threshold Distance (Optional) | The cutoff distance for the various 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, which ensures that every feature has at least one neighbor. This parameter has no effect when polygon contiguity (Contiguity edges only or Contiguity edges corners) or Get spatial weights from file spatial conceptualization is selected. | Double |

Weights Matrix File (Optional) | The path to a file containing weights that define spatial, and potentially temporal, relationships among features. | File |

Number of Neighbors
(Optional) | An integer specifying the number of neighbors to include in the analysis. | Long |

## Summary

Measures spatial autocorrelation based on feature locations and attribute values using the Global Moran's I statistic.

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, z-score, and p-value. These values are written as messages at the bottom of the Geoprocessing pane during tool execution and passed as derived output values for potential use in models or scripts. To access the messages, hover over the progress bar and click the pop-out button, or expand the details section of the messages in the Geoprocessing pane. You can also access the messages and details of a previously run tool via the geoprocessing history. Optionally, you can create an HTML report file with a graphical summary of results using this tool. The path to the report will be included with the messages summarizing the tool execution parameters. Click this path to open the report file.

Given a set of features and an associated attribute, this tool evaluates whether the pattern expressed is clustered, dispersed, or random. When the z-score or p-value 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.

- This tool calculates a z-score and p-value to indicate whether you can reject the null hypotheses. In this case, the null hypothesis states that the values of the features are spatially uncorrelated.
The z-score and p-value are measures of statistical significance. These values can help you determine whether to reject the null hypothesis. For this tool, the null hypothesis states that the values associated with features are randomly distributed.

The Input Field parameter value should contain a variety of values. The math for this statistic requires 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 the incident data. You can also use the Optimized Hot Spot Analysis tool to analyze the spatial pattern of incident data.

##### Note:

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

When the Input Feature Class parameter value is not projected (that is, when coordinates are in degrees, minutes, and seconds) or when the Output Coordinate System environment is set to a geographic coordinate system, distances will be computed using chordal measurements. Chordal distance measurements are used because they can be computed quickly and provide good estimates of true geodesic distances, at least for points within approximately 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 the two points. Chordal distances are reported in meters.

##### Caution:

Ensure that you project the data if the 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 value, if specified, should be 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 Conceptualization of Spatial Relationships parameter value 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 parameter value will ensure that 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 Distance band (sphere of influence).

- Inverse distance or Inverse distance squared
When zero is entered for the Distance Band or Threshold Distance parameter value, 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, and 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.

- FIXED_DISTANCE_BAND
The default Distance Band or Threshold Distance parameter value will ensure that 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 Distance band or threshold distance.

- INVERSE_DISTANCE or INVERSE_DISTANCE_SQUARED
When zero is entered for the Distance Band or Threshold Distance parameter value, 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, and 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.

- Fixed distance band
In Python, the derived output of this tool contains the Moran's I index value, z-score, p-value, and an HTML report file. For example, if you assign the tool's Result object to a variable named MoranResult, then MoranResult[0] stores the Moran's I index value, MoranResult[1] stores the z-score, MoranResult[2] stores the p-value, and MoranResult[3] stores the file path of the HTML report file. If you do not output an HTML report file using the Generate Report parameter, the last derived output will be an empty string.

Additional options for the Conceptualization of Spatial Relationships parameter, including three-dimensional and space-time relationships, are available using the Generate Spatial Weights Matrix tool. To use these additional options, construct a spatial weights matrix file prior to analysis; for the Conceptualization of Spatial Relationships parameter, use the Get spatial weights from file option, and for the Weights Matrix File parameter, specify the path to the spatial weights file you created.

Map layers can be specified as the Input Feature Class parameter value. When using a layer with a selection, only the selected features will be included in the analysis.

If you provide a Weights Matrix File parameter value with a .swm extension, it is expected that a spatial weights matrix file will be created using the Generate Spatial Weights Matrix tool; otherwise, an ASCII-formatted spatial weights matrix file is expected. In some cases, behavior is different depending on the following type of spatial weights matrix file you use:

- ASCII-formatted spatial weights matrix file
- Weights will be used as is. Missing feature-to-feature relationships will be treated as zeros.
- If the weights are row standardized, results may be incorrect for analyses on selection sets. To run an analysis on a selection set, convert the ASCII spatial weights file to a .swm file by reading the ASCII data into a table, and 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 will be used as is.

- ASCII-formatted spatial weights matrix file
Running an analysis with an ASCII-formatted spatial weights matrix file is memory intensive. For analyses of more than 5,000 features, consider converting the ASCII-formatted spatial weights matrix file to an SWM-formatted file. First, put the 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 value. The output will be an SWM-formatted spatial weights matrix file.

For polygon features, use the Row option for the Standardization parameter. Row standardization mitigates bias when the number of neighbors of each feature is a function of the aggregation scheme or sampling process, rather than a reflection of the actual spatial distribution of the variable you are analyzing.

For additional information about this tool's parameters, see the Modeling spatial relationships help topic.

##### Note:

It is possible to run out of memory when you run this tool. This can occur when the specified Conceptualization of Spatial Relationships and Distance Band or Threshold Distance parameter values result in features having thousands of neighbors. As a general rule, do not define spatial relationships so that features have thousands of neighbors. All features should have at least one neighbor and almost all features should have at least eight neighbors.

##### Caution:

Shapefiles cannot store null values. Tools or other procedures that create shapefiles from other types of inputs may store or interpret null values as zero. In some cases, null values are stored as large negative values in shapefiles, which can lead to unexpected results. See Geoprocessing considerations for shapefile output for more information.

## Parameters

arcpy.stats.SpatialAutocorrelation(Input_Feature_Class, Input_Field, {Generate_Report}, Conceptualization_of_Spatial_Relationships, Distance_Method, Standardization, {Distance_Band_or_Threshold_Distance}, {Weights_Matrix_File}, {number_of_neighbors})

Name | 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) | Specifies whether a graphical summary of result will be created as an .html file. - NO_REPORT—No graphical summary will be created. This is the default.
- GENERATE_REPORT—A graphical summary will be created.
| Boolean |

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—This is the 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) 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) 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) to include in the analysis 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) will be used.
- MANHATTAN_DISTANCE—The distance between two points measured along axes at right angles (city block) will be used. This is calculated by summing the (absolute) difference between the x- and y-coordinates
| String |

Standardization | Specifies whether standardization of spatial weights will be applied. 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). This is the default.
| String |

Distance_Band_or_Threshold_Distance (Optional) | The cutoff distance for the various 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, which ensures that every feature has at least one neighbor. This parameter has no effect when polygon contiguity (CONTIGUITY_EDGES_ONLY or CONTIGUITY_EDGES_CORNERS) or GET_SPATIAL_WEIGHTS_FROM_FILE spatial conceptualization is selected. | Double |

Weights_Matrix_File (Optional) | The path to a file containing weights that define spatial, and potentially temporal, relationships among features. | File |

number_of_neighbors (Optional) | An integer specifying the number of neighbors to include in the analysis. | Long |

### Derived Output

Name | Explanation | Data Type |

Index | The Moran's index value. | Double |

ZScore | The z-score. | Double |

PValue | The p-value. | Double |

Report_File | An HTML file with a graphical summary of results. | File |

### Code sample

The following Python window script demonstrates how to use the SpatialAutocorrelation function.

```
import arcpy
arcpy.env.workspace = r"c:\data"
arcpy.stats.SpatialAutocorrelation("olsResults.shp", "Residual", "NO_REPORT",
"GET_SPATIAL_WEIGHTS_FROM_FILE", "EUCLIDEAN DISTANCE",
"NONE", "#", "euclidean6Neighs.swm")
```

The following stand-alone Python script demonstrates how to use the SpatialAutocorrelation function.

```
# Analyze the growth of regional per capita incomes in U.S.
# Counties from 1969 -- 2002 using Ordinary Least Squares Regression
# Import system modules
import arcpy
# Set property to overwrite existing outputs
arcpy.env.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.env.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.stats.OrdinaryLeastSquares("USCounties.shp", "MYID",
"olsResults.shp", "GROWTH",
"LOGPCR69;SOUTH;LPCR_SOUTH;PopDen69",
"olsCoefTab.dbf",
"olsDiagTab.dbf")
# Create Spatial Weights Matrix (can be based on input or output FC)
# Process: Generate Spatial Weights Matrix...
swm = arcpy.stats.GenerateSpatialWeightsMatrix("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.stats.SpatialAutocorrelation("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 the error message.
print(arcpy.GetMessages())
```

## Environments

### Special cases

- 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. When the Output Coordinate System is based on degrees, minutes, and seconds, geodesic distances are estimated using chordal distances.

## Licensing information

- Basic: Yes
- Standard: Yes
- Advanced: Yes

#### Related topics

- An overview of the Analyzing Patterns toolset
- Modeling spatial relationships
- What is a z-score? What is a p-value?
- Find a geoprocessing tool
- Average Nearest Neighbor
- Cluster and Outlier Analysis (Anselin Local Moran's I)
- Hot Spot Analysis (Getis-Ord Gi*)
- Spatial weights
- How Spatial Autocorrelation (Global Moran's I) works