1. Clusters and Densities
Do features cluster and at what distance?
Ripley’s K (ArcMap)
Plotting Sum of Squares ®
Where are the clusters?
K-Means clustering (R)
Density surfaces
Kernel Density Methods
Hierarchical Methods
Dendrograms

2. Spatial Cluster Analysis
Wikipedia

3. Ripley’s K Measure of spatial homogeneity

4. L(d) Expected: for random data Observed: k drives the model
k = 1 when distance from i to j < d
k = 0 when distance from i to j >=d

5. Ripley’s K
Counts the number of points within a distance of each other

6. Cluster Analysis
ArcGIS Help

7. Ripley’s K - ArcGIS
ArcGIS Help

8. Ripley’s K
Small circles have no points within distance

9. Ripley’s K
At the right distance, we have more points within the distance than expected

10. Ripley’s K
At large distances, we include the expected number of points.

11. Random Data

12. Multi-Distance Cluster Analysis