# 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.

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## 3 Answers

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

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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
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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.

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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
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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)

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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
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