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:

enter image description here

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:

enter image description here

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.

My questions:

  1. Are you familiar with common measure of dispersion that take into account multiple clusters?
  2. What is your opinion about the proposed measure?
  3. 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?
  • $\begingroup$ Essentially, I think that what I am looking for is a way to measure the dispersion of a multi-modal distribution.. $\endgroup$ – edelburg Jul 5 '18 at 14:36

As it turns out, Entropy makes a good measure at least for the purpose I described. I use it like so: each cluster has its own probability computed as the ratio between the points it includes and the total number of points. With these probabilities, the entropy can be computed as follows: -sum(pi*log(pi)). Lower entropy implies more "clusteredness" and higher entropy implies more dispersion.


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