  This paper investigates the model degrees of freedom in k-means clustering.
An extension of Stein's lemma provides an expression for the effective degrees
of freedom in the k-means model. Approximating the degrees of freedom in
practice requires simplifications of this expression, however empirical studies
evince the appropriateness of our proposed approach. The practical relevance of
this new degrees of freedom formulation for k-means is demonstrated through
model selection using the Bayesian Information Criterion. The reliability of
this method is validated through experiments on simulated data as well as on a
large collection of publicly available benchmark data sets from diverse
application areas. Comparisons with popular existing techniques indicate that
this approach is extremely competitive for selecting high quality clustering
solutions. Code to implement the proposed approach is available in the form of
an R package from https://github.com/DavidHofmeyr/edfkmeans.
