# Clustering (k-means, or otherwise) with a minimum cluster size constraint

I need to cluster units into $k$ clusters to minimize within-group sum of squares (WSS), but I need to ensure that the clusters each contain at least $m$ units. Any idea if any of R's clustering functions allow for clustering into $k$ clusters subject to a minimum cluster size constraint? kmeans() does not seem to offer a size constraint option.

Use EM Clustering

In EM clustering, the algorithm iteratively refines an initial cluster model to fit the data and determines the probability that a data point exists in a cluster. The algorithm ends the process when the probabilistic model fits the data. The function used to determine the fit is the log-likelihood of the data given the model.

If empty clusters are generated during the process, or if the membership of one or more of the clusters falls below a given threshold, the clusters with low populations are reseeded at new points and the EM algorithm is rerun.

• Thanks, Marianna. I would prefer a solution that relies less heavily on (typically, unjustifiable) parametric models, but will definitely look into it. – Cyrus S Dec 12 '10 at 15:59

This problem is addressed in this paper:

Bradley, P. S., K. P. Bennett, and Ayhan Demiriz. "Constrained k-means clustering." Microsoft Research, Redmond (2000): 1-8.

I have an implementation of the algorithm in python.

• This is perfect, thanks! I used the rPython package in R to create an interface to this implementation that I accessed from my R script. – Michael Ohlrogge Feb 27 '17 at 19:57
• @MichaelOhlrogge do you have an example somewhere (github?) on the interface you wrote to call that python package form R? Thanks! – Matifou Feb 8 at 23:53
• Sorry, I looked around my old code but couldn't find it anymore. – Michael Ohlrogge Feb 9 at 1:23

I think it would just be a matter of running the k means as part of an if loop with a test for cluster sizes, I.e. Count n in cluster k - also remember that k means will give different results for each run on the same data so you should probably be running it as part of a loop anyway to extract the "best" result

• Thanks, Alex. I see a problem with this though: what if over the loops the solutions generated never satisfy the constraint? That could happen if k means were set to run with no cluster size constraint. I'd love a solution that avoids this. (The nature of the application is such that I really need to ensure clusters are of a minimum size.) – Cyrus S Dec 10 '10 at 21:50

How large is your data set? Maybe you could try to run a hierarchical clustering and then decide which clusters retain based on your dendrogram.

If your data set is huge, you could also combine both clustering methods: an initial non-hierarchical clustering and then a hierarchical clustering using the groups from the non-hierarchical analysis. You can find an example of this approach in Martínez-Pastor et al (2005)

• Thanks, Manuel. This actually sounds like a very intriguing possibility. I need to think about whether the hierarchical partitioning would impose certain constraints that would prevent the algorithm from achieving the optimal cluster partitioning directly under the size constraint. But intuitively, I can see that this might work. – Cyrus S Dec 12 '10 at 16:01

This can be achieved by modifying the cluster assignment step (E in EM) by formulating it as a Minimum Cost Flow (MCF) linear network optimisation problem.

I have written a python package which uses Google's Operations Research tools's SimpleMinCostFlow which is a fast C++ implementation. Its has a standard scikit-lean API.