The use of a common location measure in the invariant coordinate selection and projection pursuit

Invariant coordinate selection (ICS) and projection pursuit (PP) are two methods that can be used to detect clustering directions in multivariate data by optimizing criteria sensitive to non-normality. In particular, ICS finds clustering directions using a relative eigen-decomposition of two scatter matrices with different levels of robustness; PP is a one-dimensional variant of ICS. Each of the two scatter matrices includes an implicit or explicit choice of location. However, when different measures of location are used, ICS and PP can behave counter-intuitively. In this paper we explore this behavior in a variety of examples and propose a simple and natural solution: use the same measure of location for both scatter matrices.