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Procrustes Problems$
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John C Gower and Garmt B Dijksterhuis

Print publication date: 2004

Print ISBN-13: 9780198510581

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198510581.001.0001

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Weighting, scaling, and missing values

Weighting, scaling, and missing values

(p.91) 8 Weighting, scaling, and missing values
Procrustes Problems

J. C. Gower

G. B. Dijksterhuis

Oxford University Press

This chapter introduces the different forms of weighting. One form of weighting is when the rows of X1 are weighted. It is also possible to weight the columns. In these cases the weighting is expressed in terms of diagonal matrices. The most general form of weighting is when every cell of X1 gets separate weighting. Missing values may be specified by giving rows, columns, or cells zero weights. The important case of isotropic scaling is considered, where a scaling factor can be applied to the whole of the matrix X1 . This allows for the common situation where the relative sizes of X1 and X2 are unknown. Anisotropic scaling is introduced, represented by diagonal matrices S and T. Unlike weighting-matrices, which are given, scaling matrices need to be estimated. However, in iterative algorithms, the current estimates of scaling matrices may be treated as given weights while estimating updates of transformation matrices or further scaling matrices.

Keywords:   weighting, anisotropic scaling, isotropic scaling, transformation matrices

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