Context: My data are binary maps that represent the spatial distribution of events. Whenever an event occurs at a location, the pixel representing that location is assigned with the value 1. Below are two examples of such binary maps:
The problem: The points are clustered by distance. E.g., for a certain distance, the left map above has 2 clusters, each consists of 6 points. The right map has five clusters of size: 6, 6, 3, 1 and 1. I wish to measure the dispersion of the points in each maps.
What I have this far I know there exist ways to measure spatial dispersion based on variance, covariance, etc. However, these are useful for measuring the dispersion of points in a single cluster. I came up with a simple measure for dispersion of multiple clusters of points, which accounts for the number of clusters and their size:
where Ni is the number of points in the i-th cluster. This sum produces small values for very clustered data and larger values for more dispersed data. The sum increases as more clusters occur, taking their size into account: a large cluster contributes a small amount to the sum and vice versa. A cluster with a single point will contribute the largest value to the sum, 1.
E.g., the measure for the left map is c=(1/6)+(1/6)=0.33 and for the right map it is c=(1/6)+(1/6)+(1/3)+1+1=3. Indeed, the points in the left map are packed into a smaller number of clusters than points in the right map.
- Are you familiar with common measure of dispersion that take into account multiple clusters?
- What is your opinion about the proposed measure?
- I wish to incorporate the spatial variance within each cluster as a third factor that affects dispersion. Do you know a measure that also takes inter-cluster variance into account? Do you have an idea on how to incorporate this information into the proposed measure?