  Regression mixture models are widely studied in statistics, machine learning
and data analysis. Fitting regression mixtures is challenging and is usually
performed by maximum likelihood by using the expectation-maximization (EM)
algorithm. However, it is well-known that the initialization is crucial for EM.
If the initialization is inappropriately performed, the EM algorithm may lead
to unsatisfactory results. The EM algorithm also requires the number of
clusters to be given a priori; the problem of selecting the number of mixture
components requires using model selection criteria to choose one from a set of
pre-estimated candidate models. We propose a new fully unsupervised algorithm
to learn regression mixture models with unknown number of components. The
developed unsupervised learning approach consists in a penalized maximum
likelihood estimation carried out by a robust expectation-maximization (EM)
algorithm for fitting polynomial, spline and B-spline regressions mixtures. The
proposed learning approach is fully unsupervised: 1) it simultaneously infers
the model parameters and the optimal number of the regression mixture
components from the data as the learning proceeds, rather than in a two-fold
scheme as in standard model-based clustering using afterward model selection
criteria, and 2) it does not require accurate initialization unlike the
standard EM for regression mixtures. The developed approach is applied to curve
clustering problems. Numerical experiments on simulated data show that the
proposed robust EM algorithm performs well and provides accurate results in
terms of robustness with regard initialization and retrieving the optimal
partition with the actual number of clusters. An application to real data in
the framework of functional data clustering, confirms the benefit of the
proposed approach for practical applications.
