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00925njm 22002777a 4500 dc Cerioli, Andrea author García Escudero, Luis Ángel author Mayo Iscar, Agustín author Riani, Marco author 2018 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. Journal of Computational and Graphical Statistics, 2016, vol. 27, p. 404-416 1061-8600 http://uvadoc.uva.es/handle/10324/32023 10.1080/10618600.2017.1390469 1537-2715 Finding the number of normal groups in model-based clustering via constrained likelihoods