A Projection Pursuit Algorithm for Preference Data

In the framework of preference rankings, the interest can lie in finding which predictors and which interactions are able to explain the observed preference structures. The last years have seen a remarkable owering of works about the use of decision tree for clustering preference vectors. As a matter of fact, decision trees are useful and intuitive, but they are very unstable: small perturbations bring big changes. This is the reason why it could be necessary to use more stable procedures in order to clustering ranking data. In this work, following the idea of Bolton (2003), a Projection Pursuit (PP) clustering algorithm for preference data will be proposed in order to extract useful information in a low-dimensional subspace by starting from a high but most empty dimensional space. Projection pursuit clustering is a synthesis of projection pursuit and nonhierarchical clustering methods that simultaneously attempts to cluster the data and to find a low-dimensional representation of this cluster structure. As introduced by Huber (1985), a PP algorithm consists of two components: an index function I(α) that measures the "usefulness" of projection and a search algorithm that varies the projection direction so as to find the optimal projections, given the index function I(α) and the data set X. In this work a proper specified Projection index function for discrete data will be defined: several distances will be used to evaluate distances between the density of the projected data and the uninteresting uniform density. We also propose diagnostics for finding the optimum number of clusters in projection pursuit clustering. All the methodology is illustrated and evaluated on one simulated and one real dataset.