Decision boundaries for mixtures of regressions

The analysis of the decision boundaries plays an important role in understanding the characteristics of a classifier in the framework of model-based clustering and discriminant analysis. The wider is the family of decision boundaries generated by a classifier the larger is its flexibility for classification purposes. In this paper, we present rigorous results concerning the decision boundaries of mixtures of (linear) regressions under Gaussian assumptions. In particular, three types of mixtures of regressions are considered: with fixed covariates, with concomitant variables, and with random covariates. The obtained decision boundaries have a geometrical interpretation in terms hyperquadrics and define a taxonomy of the considered models. Beyond Gaussian assumptions, decision boundaries can be investigated numerically; as an example, we illustrate the case of the tt distribution.