# Chosing optimal k and optimal distance-metric for k-means [duplicate]

I have a data-set with roughly 20-dimensions and millions of points which I want to cluster.

The goal is to find a set of clusters which:

• Are as distinct as possible from each other (minimum coupling) between clusters, i.e. maximum average distance between centroids
• With instances as similar as possible (maximum cohesion) inside the clusters, i.e. minimum average distance between the points within each cluster.

Without considering the domain, is there a good metric to help determine the optimal $k$ I should choose?

Intuitively, I would pick $k=\sqrt{N}$ for a data-set in two dimensions, and $k=\sqrt[M]{N}$ for a data-set with $M$ dimensions and $N$ data-points, but I have a hunch that there are better methods.

A related and complementary question is which distance metric to use. Due to the curse of dimensionality, I know that euclidean distance becomes a poor choice as the number of dimensions increases.

Note that this question is different than Choosing optimal K for KNN (this one asks about clustering rather than k-NN classification)

## 1 Answer

There is plenty of literature on choosing k, such as the silhouette plot. This question has also been asked a dozen times.

Same for your second question - asked several times before:

Do not use k-means with other distance functions than sum-of-squares. It may stop converging. k-means is not distance based. It minimizes the very classic sum of squares. The mean function is an L2 estimator of centrality - if you want to use a different distance function, you need to choose cluster centers differently. This has been done, see k-medoids aka PAM.

Don't forget to spend a lot of time preprocessing your data and visualizing your results. People tend to neglect that, and get really bad results without noticing... again, see other questions here on k-means.

• Thanks for the response. I found several related questions by searching for the kmeans tag. Some appear on the right as 'Related' but none of them answered these specific questions in a clear direct way. I now realize I should have asked about distance clustering rather than specifically about k-means. I'll look up Silhouette plots, thanks again. – arielf - Reinstate Monica Apr 16 '15 at 6:58