Maximum Likelihood Estimation of Gaussian Cluster Weighted Models and Relationships with Mixtures of Regression

TL;DR

This paper analyzes Gaussian cluster-weighted models using maximum likelihood estimation and reveals their relationship with mixtures of regression, showing that under certain conditions, they produce equivalent parameter estimates.

Contribution

It demonstrates that Gaussian CWMs are nested within finite mixtures of regression models and provides a detailed likelihood analysis under Gaussian assumptions.

Findings

Gaussian CWMs and mixtures of regression share the same estimates under certain conditions

The likelihood function analysis clarifies the relationship between these models

CWMs can be viewed as a generalization of regression mixture models

Abstract

Cluster-weighted modeling (CWM) is a mixture approach for modeling the joint probability of a response variable and a set of explanatory variables. The parameters are estimated by means of the expectation-maximization algorithm according to the maximum likelihood approach. Under Gaussian assumptions, we analyse the complete-data likelihood function of cluster weighted models. Further, under suitable hypotheses we show that the maximization of the likelihood function of Gaussian cluster weighted models leads to the same parameter estimates of finite mixtures of regression and finite mixtures of regression with concomitant variables. In this sense, the latter ones can be considered as nested models of Gaussian cluster weighted models.