Invariant Coordinate Selection

A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based upon the eigenvalue-eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant coordinate system for the multivariate data. Consequently, we view this method as a method for invariant coordinate selection (ICS). By plotting the data with respect to this new invariant coordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant coordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant coordinates corresponds to Fisher’s linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.

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