# 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.. – geotheory Dec 16 '16 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.. – Michael Chernick Dec 16 '16 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? – geotheory Dec 16 '16 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. – ttnphns Dec 17 '16 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). – geotheory Dec 17 '16 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 – geotheory Dec 17 '16 at 11:17