Decision boundaries for mixtures of regressions

Abstract

a b s t r a c tThe 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 t distribution.

Multivariate Response and Parsimony for Gaussian Cluster-Weighted Models

“…Hennig, 2000), i.e., the assignment of an observation to a cluster is also dependent on the distribution of the covariates. In such models, the component covariate distributions can be distinct; for a discussion about the difference between FMR, FMRC, and CWMs from a geometrical point of view, see Ingrassia and Punzo (2016). Some extensions of this methodology have dealt with non-linear local relationships (Punzo, 2014), high dimensional covariates (Subedi et al, 2013(Subedi et al, , 2015, and various response types (Ingrassia et al, 2015;Punzo andIngrassia, 2013, 2015).…”

A new look at the inverse Gaussian distribution with applications to insurance and economic data

“…More generally, by solving (17) as a function of x, the positive real line is partitioned in two regions, say R good and R bad , of good and bad data, respectively. By using the (monotonically increasing) logarithmic transformation of the discriminant functions, as in Ingrassia and Punzo [43], the inequality in (17) becomes the following quadratic inequality…”

Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers

“…The upper linear cluster represents an area of good quality soil, characterized by flat land, with soil rich in nutrients and where silvicultural treatments have enabled the trees to be free of competition and expose themselves to light, which creates a greater growth in diameter and height. Because the true group-membership is not available, an N-MRM with 3 mixture components and assuming a Gaussian distribution for the diameter in each group (see, e.g., Ingrassia et al, 2015, Ingrassia and Punzo, 2016, is estimated on the data in Figure 2 …”