Deciding the number of clusters k is one of the most difficult problems in clus-
ter analysis. For this purpose, complexity-penalized likelihood approaches have been
introduced in model-based clustering, such as the well known BIC and ICL crite-
ria. However, the classi cation/mixture likelihoods considered in these approaches
are unbounded without any constraint on the cluster scatter matrices. Constraints
also prevent traditional EM and CEM algorithms from being trapped in (spurious)
local maxima. Controlling the maximal ratio between the eigenvalues of the scatter
matrices to be smaller than a xed constant c 1 is a sensible idea for setting such
constraints. A new penalized likelihood criterion which takes into account the higher
model complexity that a higher value of c entails, is proposed. Based on this criterion,
a novel and fully automated procedure, leading to a small ranked list of optimal (k; c)
couples is provided. A new plot called \car-bike" which provides a concise summary
of the solutions is introduced. The performance of the procedure is assessed both in
empirical examples and through a simulation study as a function of cluster overlap.
Supplemental materials for the article are available online.
