MultiDistance Spatial Cluster Analysis (Ripley's kfunction) (Spatial Statistics) 

Release 9.2
Last modified January 9, 2009 
Print all topics in : "Tools" 
The MultiDistance Spatial Cluster Analysis (Ripley's Kfunction) tool determines whether a feature class is clustered at multiple different distances. The tool outputs the result as a table and optionally as a pop up graphic.
Learn more about how MultiDistance Spatial Cluster Analysis works.
Usage tips
The output of the tool is a table with two fields named "ExpectedK" and "ObservedK" containing the expected k and observed k values respectively. If a confidence interval option is specified two additional fields named "LowConfEnv" and "HiConfEnv" will be present with the confidence interval information for each iteration of the tool.
When the "Display Output Graphically" option is chosen, a graph showing the expected and observed outputs for each iteration is generated and displayed. The expected results will be represented by a blue line while the observed results will be a red line. Deviation of the observed line above the expected line indicates that the dataset is exhibiting clustering at that distance. Deviation of the observed line below the expected line indicates that the dataset is exhibiting dispersion at that distance.
The Weight Field is most appropriately used when it represents number of incidents or counts.
When no weight field is specified, the confidence envelope is constructed by distributing points randomly in the study area and calculating k for that distribution. Each random distribution of the points is called a "permutation". If "99 permutations" is selected, the tool will randomly distribute the set of points 99 times for each iteration. After distributing the points 99 times the tool selects the k value that deviated above and below the expected by the greatest amount and these values become the confidence interval.
When a weight field is specified, only the weight values are randomly redistributed to compute confidence envelopes; the point locations remain fixed. In essence, when a weight field is specified, locations remain fixed and we evaluate the clustering of feature values in space. On the other hand, when no weight field is specified we are analyzing clustering/dispersion of feature locations.
When no study area is specified, the tool uses a minimum enclosing rectangle as the study area polygon.
The kfunction statistic is very sensitive to the size of the study area. Identical arrangments of points can exhibit clustering or dispersion depending on the size of the study area. Therefore it is imperative that the study area boundaries are carefully considered. If no study area feature class is provided, the minimum bounding rectangle of the input features is used. The picture below is a classic example of how identical feature distributions can be dispersed or clustered depending on the area specified.
A study area feature class should only be given if "Userprovided Study Area Feature Class" is chosen for the Study Area Method parameter.
If a study area feature class is specified, it should have exactly one single part feature.
If no Beginning Distance or Increment Distance are specified then default values are calculated for you based on the extent of the input feature class.
Points in the input feature class that fall outside the user specified study area are only considered when the "None" edge correction option is selected. Other edge correction techniques compensate for edge issues with simulated points, by reducing the study area, or by weighting edge neighbors higher than nonedge neighbors.
The Simulate Outer Boundary Values edge correction method mirrors points across the study area boundary to correct for underestimates near edges. Points that are within a distance equal to the maximum distance band of the edge of the study area are mirrored. The mirrored points are used so that edge points will have more accurate neighbor estimates. The diagram below illustrates what points will be used in the calculation and which will be used only for edge correction.
The Reduce Analysis Area edge correction technique shrinks the size of the analysis area by a distance equal to the largest distance band to be used in the analysis. After shrinking the study area, points found outside of the new study area will be considered only when neighbor counts are being assessed for points still inside the study area. They will not be used in any other way during the kfunction calculation. The diagram below illustrates what points will be used in the calculation and which will be used only for edge correction.
Ripley's Edge Correction Formula checks each point's distance from the edge of the study area and its distance to each of its neighbors. All neighbors that are further away from the point in question than the edge of the study area are given extra weight. This edge correction method is only appropriate for square or rectangular shaped study areas.
Mathematically, the MultiDistance Spatial Cluster Analysis tool uses a common transformation of Ripley's kfunction where the expected result with a random set of points is equal to the input distance. The transformation L(d) is shown below.
where A is area, N is the number of points, d is the distance and k(i, j) is the weight, which (if there is no edge correction) is 1 when the distance between i and j is less than or equal to d and 0 when the distance between i and j is greater than d. When edge correction is applied, the weight of k(i,j) is modified slightly.
The units of the "Beginning Distance" and "Distance Increment" are the units of the input Feature Class' coordinate system.
For line and polygon features, feature centroids are used in computations.
The "Display Output Graphically" parameter will only work on the windows operating system. When set to true it will display the results of the tool graphically.
The environment settings do not have an effect on this tool.
Command line syntax
An overview of the Command Line window
MultiDistanceSpatialClustering_stats <Input_Feature_Class> <Output_Table> <Number_of_Distance_Bands> {0 Permutations  no confidence envelope  9 Permutations  99 Permutations  999 Permutations} {Display_Results_Graphically} {Weight_Field} {Beginning_Distance} {Distance_Increment} {None  Simulate Outer Boundary Values  Reduce Analysis Area  Ripley's Edge Correction Formula} {Minimum Enclosing Rectangle  User provided Study Area Feature Class} {Study_Area_Feature_Class}
Parameter  Explanation  Data Type 
<Input_Feature_Class> 
The feature class upon which the analysis will be performed.

Feature Class 
<Output_Table> 
The table to which the results of the analysis will be written.

Table 
<Number_of_Distance_Bands> 
The number of times to increment the neighborhood size and analyze the dataset for clustering. The starting point and size of the increment are specified in the Beginning Distance and Distance Increment parameters respectively.

Long 
{0 Permutations  no confidence envelope  9 Permutations  99 Permutations  999 Permutations} 
The confidence envelope is calculated by randomly placing points in the study area. The number of points randomly placed is equal to the number of points in the feature class. Each set of random placements is called a "permutation" and the confidence envelope is created from these permutations. This parameter allows you to select how many permutations you want to use to create the confidence envelope.

String 
{Display_Results_Graphically} 
Specifies whether the tool will display the results of the MultiDistance Spatial Cluster Analysis tool graphically.

Boolean 
{Weight_Field} 
A numeric field with weights that give certain features more influence than others.

Field 
{Beginning_Distance} 
The distance at which to start the cluster analysis and the distance from which to increment. The value entered for this parameter should be in the units of the Input Feature Class' coordinate system.

Double 
{Distance_Increment} 
The distance by which to increment during each iteration. The distance used in the analysis starts at the Beginning Distance and increments by the amount specified in the Distance Increment. The value entered for this parameter should be in the units of the Input Feature Class' coordinate system.

Long 
{None  Simulate Outer Boundary Values  Reduce Analysis Area  Ripley's Edge Correction Formula} 
Method to use to correct for under estimates in the number of neighbors for features near the edges of the study area.

String 
{Minimum Enclosing Rectangle  User provided Study Area Feature Class} 
Specifies the region to use for the study area. Selection of this area is critical as area is part of the equation used by the tool.

String 
{Study_Area_Feature_Class} 
Feature class that delineates the area over which the input feature class should be analyzed. Only to be specified if Userprovided Study Area Feature Class is specified in the Study Area Feature Class parameter.

Feature Class 
Scripting syntax
About getting started with writing geoprocessing scripts
MultiDistanceSpatialClustering_stats (Input_Feature_Class, Output_Table, Number_of_Distance_Bands, Compute_Confidence_Envelope, Display_Results_Graphically, Weight_Field, Beginning_Distance, Distance_Increment, Boundary_Correction_Method, Study_Area_Method, Study_Area_Feature_Class)
Parameter  Explanation  Data Type 
Input_Feature_Class (Required) 
The feature class upon which the analysis will be performed.

Feature Class 
Output_Table (Required) 
The table to which the results of the analysis will be written.

Table 
Number_of_Distance_Bands (Required) 
The number of times to increment the neighborhood size and analyze the dataset for clustering. The starting point and size of the increment are specified in the Beginning Distance and Distance Increment parameters respectively.

Long 
Compute_Confidence_Envelope (Optional) 
The confidence envelope is calculated by randomly placing points in the study area. The number of points randomly placed is equal to the number of points in the feature class. Each set of random placements is called a "permutation" and the confidence envelope is created from these permutations. This parameter allows you to select how many permutations you want to use to create the confidence envelope.

String 
Display_Results_Graphically (Optional) 
Specifies whether the tool will display the results of the MultiDistance Spatial Cluster Analysis tool graphically.

Boolean 
Weight_Field (Optional) 
A numeric field with weights that give certain features more influence than others.

Field 
Beginning_Distance (Optional) 
The distance at which to start the cluster analysis and the distance from which to increment. The value entered for this parameter should be in the units of the Input Feature Class' coordinate system.

Double 
Distance_Increment (Optional) 
The distance by which to increment during each iteration. The distance used in the analysis starts at the Beginning Distance and increments by the amount specified in the Distance Increment. The value entered for this parameter should be in the units of the Input Feature Class' coordinate system.

Long 
Boundary_Correction_Method (Optional) 
Method to use to correct for under estimates in the number of neighbors for features near the edges of the study area.

String 
Study_Area_Method (Optional) 
Specifies the region to use for the study area. Selection of this area is critical as area is part of the equation used by the tool.

String 
Study_Area_Feature_Class (Optional) 
Feature class that delineates the area over which the input feature class should be analyzed. Only to be specified if Userprovided Study Area Feature Class is specified in the Study Area Feature Class parameter.

Feature Class 