On Invariant Coordinate System (ICS) Functionals |
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| Authors: | Pauliina Ilmonen Hannu Oja Robert Serfling |
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| Affiliation: | 1. School of Health Sciences, FI‐33014 University of Tampere, Finland, and Aalto University School of Economics, PL 21210, FI‐00076 Aalto, Finland;2. School of Health Sciences, FI‐33014 University of Tampere, Finland;3. Department of Mathematics, University of Texas at Dallas, Richardson, TX 75080‐3021, USA E‐mail: pauliina.ilmonen@gmail.com |
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| Abstract: | 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 (ICS). In this paper, standardization of multivariate distributions and characteristics of 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 & Sengupta (1993) are examined. Diverse applications of ICS functionals are discussed. |
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| Keywords: | Invariant coordinate system functionals multivariate analysis nonparametric methods |
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