On Invariant Coordinate System (ICS) Functionals

Equivariance and invariance issues often arise in multivariate statistical analysis. Statistical procedures have to be modified sometimes to obtain an affine equivariant or invariant version. This is often done by preprocessing the data, e.g., by standardizing the multivariate data or by transforming the data to an invariant coordinate system. In this article standardization of multivariate distributions, and characteristics of invariant coordinate system (ICS) functionals and statistics, are examined. Also, invariances up to some groups of transformations are discussed. Constructions of ICS functionals are addressed. In particular the construction based on the use of two scatter matrix functionals presented by Tyler et al. (2009), and direct definitions based on the approach presented by Chaudhuri and Sengupta (1993) are examined. Diverse applications of ICS functionals are discussed.