  In model-based clustering and classification, the cluster-weighted model
constitutes a convenient approach when the random vector of interest
constitutes a response variable Y and a set p of explanatory variables X.
However, its applicability may be limited when p is high. To overcome this
problem, this paper assumes a latent factor structure for X in each mixture
component. This leads to the cluster-weighted factor analyzers (CWFA) model. By
imposing constraints on the variance of Y and the covariance matrix of X, 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 the
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.
