Directed Clustering of Multivariate Data Based on Linear or Quadratic Latent Variable Models

We consider situations in which the clustering of some multivariate data is desired, which establishes an ordering of the clusters with respect to an underlying latent variable. As our motivating example for a situation where such a technique is desirable, we consider scatterplots of traffic flow and speed, where a pattern of consecutive clusters can be thought to be linked by a latent variable, which is interpretable as traffic density. We focus on latent structures of linear or quadratic shapes, and present an estimation methodology based on expectation-maximization, which estimates both the latent subspace and the clusters along it. The directed clustering approach is summarized in two algorithms and applied to the traffic example outlined. Connections to related methodology, including principal curves, are briefly drawn.