Clustering and classification via cluster-weighted factor analyzers

In model-based clustering and classification, the cluster-weighted model is a convenient approach when the random vector of interest is constituted by a response variable <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$Y$$</EquationSource> </InlineEquation> and by a vector <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\varvec{X}}$$</EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation> covariates. However, its applicability may be limited when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation> is high. To overcome this problem, this paper assumes a latent factor structure for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${\varvec{X}}$$</EquationSource> </InlineEquation> in each mixture component, under Gaussian assumptions. This leads to the cluster-weighted factor analyzers (CWFA) model. By imposing constraints on the variance of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$Y$$</EquationSource> </InlineEquation> and the covariance matrix of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${\varvec{X}}$$</EquationSource> </InlineEquation>, a novel family of sixteen CWFA models is introduced for model-based clustering and classification. The alternating expectation-conditional maximization algorithm, for maximum likelihood estimation of the parameters of all models in the family, is described; to initialize the algorithm, a 5-step hierarchical procedure is proposed, which uses the nested structures of the models within the family and thus guarantees the natural ranking among the sixteen likelihoods. Artificial and real data show that these models have very good clustering and classification performance and that the algorithm is able to recover the parameters very well. Copyright Springer-Verlag Berlin Heidelberg 2013