A1 Refereed original research article in a scientific journal
On Invariant Coordinate System (ICS) Functionals
Authors: Ilmonen P, Oja H, Serfling R
Publisher: WILEY-BLACKWELL
Publication year: 2012
Journal: International Statistical Review
Journal name in source: INTERNATIONAL STATISTICAL REVIEW
Journal acronym: INT STAT REV
Volume: 80
Issue: 1
First page : 93
Last page: 110
Number of pages: 18
ISSN: 0306-7734
DOI: https://doi.org/10.1111/j.1751-5823.2011.00163.x
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.
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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