# Measures of clustering vs dispersal

I'm looking for metrics of the degree of grouping/clustering of spatial data. I'm not looking to formally cluster the data i.e. classify points within groupings. But rather an index such as from 0 for a uniform distribution to 1 for a set of spatially coinciding points. All metrics I've seen so far either relate to distributions from the mean or validation of classified clusters. I need something agnostic from any formal groupings, and which works with 2d data (although I'm guessing the solution could also work with 1d data). Grateful for any pointers.

• I wonder if I'm being sucked into interval analysis.. Dec 16, 2016 at 11:01
• Number of clusters picked and how many members go in each group do depend on the variation within clusters and the variation between clusters, probably based on cluster centers. All of the concepts require a choice of an appropriate distance measure.. Dec 16, 2016 at 13:16
• @MichaelChernick that sounds like the formal clustering approach that I'm hoping to avoid. Surely a metric is possible based purely on distances between all points to all others? Dec 16, 2016 at 19:51
• I need something agnostic from any formal groupings and looking for metrics of the degree of grouping/clustering look contradictive. There are many clustering algorithms each defining what is "cluster" formally/conceptually differently. Therefore, they imply the many different "metrics of grouping". Honestly, to assess if there any clear groups one has first to try to cluster the data into groups. Even our eye, it discerns groups of points on a plot because it has convincingly clumped them. Dec 17, 2016 at 14:03
• It's not contradictory @ttnphns. Just relax your conceptualisation - points can be spatially "clustered" without being formally classified as such. I'm looking for methods that do not involve "clustering" the data (with for example k-means). Dec 17, 2016 at 17:24

There is e.g. Hopkins statistic of clustering tendency.

But in my opinion these statistics are useless. Let me explain.

Essentially, these statistics usually are a very weak hypothesis test, with H0 being "my data is uniform distributed".

But a data may be clearly not uniform, but nevertheless not "clustering" in the usual sense. For example, a single Gaussian distribution will fail these tests, but not cluster with any algorithm. Any data set can be made "clustered" by applying exp to one attribute.

The assumption not uniform $\Rightarrow$ clustered is too simple.

What you are really interested in is whether the data is multi modal rather than unimodal. A better test would therefore try to prove unimodality. But I don't know any.

• Thanks @Anony-Mousse most useful. I'll continue racking my brains on a logical approach to this Dec 17, 2016 at 11:17
• I think that "useless" is too strong here. They are useless for detecting multimodality, as you pointed out, but not for detecting whether data points will typically be closer together than would be expected for a uniform distribution. It's just a different problem. Oct 19, 2022 at 17:36