Tandem clustering with invariant coordinate selection

For multivariate data with noise variables, tandem clustering is a well-known technique that aims to improve cluster identification by first reducing the dimension. However, the usual approach using principal component analysis (PCA) has been criticized for focusing only on inertia so that the first components do not necessarily retain the structure of interest for clustering. To overcome this drawback, a new tandem clustering approach based on invariant coordinate selection (ICS) is proposed. By jointly diagonalizing two scatter matrices, ICS is designed to find structure in the data while returning affine invariant components. Some theoretical results have already been derived and guarantee that under some elliptical mixture models, the group structure can be highlighted on a subset of the first and/or last components. Nevertheless, ICS has received little attention in a clustering context. Two challenges are the choice of the pair of scatter matrices and the selection of the components to retain. For clustering purposes, it is demonstrated that the best scatter pairs consist of one scatter matrix that captures the within-cluster structure and another that captures the global structure. For the former, local shape or pairwise scatters are of great interest, as is the minimum covariance determinant (MCD) estimator based on a carefully selected subset size that is smaller than usual. The performance of ICS as a dimension reduction method is evaluated in terms of preserving the cluster structure present in data. In an extensive simulation study and in empirical applications with benchmark data sets, different combinations of scatter matrices as well as component selection criteria are compared in situations with and without outliers. Overall, the new approach of tandem clustering with ICS shows promising results and clearly outperforms the approach with PCA.