ICS with positive semi-definite scatter matrices for data not in general position

Abstract

The Invariant Coordinate Selection method is aimed at detecting some potential structure of groups or the presence of outliers in a multivariate data base and in an unsupervised way. ICS is based on the joint diagonalisation of two scatter matrices which are assumed to be non-singular. In practice, however, because of some multicollinearity problems in high dimension, the scatter matrices may be singular. In such a context, it is possible to generalize ICS by using some Generalized Singular Value Decomposition. This approach has some advantages in particular compared with another approach based on generalized inverse of scatter matrices. In some examples where the structure of interest is contained in some subspace, the proposed method is able to recover the subspace of interest while other approaches may fail in identifying such a subspace. These advantages are discussed in detail from a theoretical point of view and using some simulated examples.