We review the class of discrete latent variable models, and we propose a new formulation of the hidden Markov model for compositional data. We illustrate some results of the analysis of the expenditures of the Spanish regions over several decades, showing that the approach is promising to cluster regions with different patterns linked to the composition of parts in the system over time. We give particular emphasis to the possible developments of discrete latent variable models that take inspiration from common problems of these models, such as the multimodality of the likelihood function and issues related to the choice of the number of support points of the latent variables.