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Least Squares Approximation of Matrices by Additive and Multiplicative Models

Gabriel, K. R.·1978·Journal of the Royal Statistical Society Series B: Statistical Methodology
DOI 10.1111/j.2517-6161.1978.tb01663.x
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Abstract

Summary: Reduced rank approximation of matrices by Householder–Young methods is shown to be equivalent to fitting of a bilinear (multiplicative) model and to projection onto an optimally chosen subspace. The relation to linear approximation is made evident. Least squares fitting of a mixed linear (additive) and bilinear (multiplicative) model is proved to be a two-stage process: (1) fit the linear part of the model, then take residuals, and (2) fit the bilinear part to the residuals. Extensions of this result are given as well as some results on distributions of residuals and tests against alternatives of given rank. Statistical applications are shown in principal component analysis, in biplot graphical display and in fitting additive or Mandel-type models to two-way tables.

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Research works printed but not linked: 32

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  2. Bradu, 1978, The biplot as a diagnostic tool for models in two-way tables, Technometrics, 20, 47, 10.1080/00401706.1978.10489617 · DOI · Paper page
  3. Corsten, 1972, Multiplicative effects in two-way analysis of variance, Statist. Neerlandica, 26, 61, 10.1111/j.1467-9574.1972.tb00173.x · doi:10.1111/j.1467-9574.1972.tb00173.x
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  8. Gabriel, 1972, Analysis of meterological data by means of canonical decomposition and biplots, J. Appl. Meteor., 11, 1071, 10.1175/1520-0450(1972)011<1071:AOMDBM>2.0.CO;2 · doi:10.1175/1520-0450(1972)011<1071:aomdbm>2.0.co;2
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Metadata

Authors
Gabriel, K. R.
Venue
Journal of the Royal Statistical Society Series B: Statistical Methodology
Publisher
Oxford University Press (OUP)
Published
1978-01-01
Volume
40
Issue
2
Pages
186-196
ISSN
  • 1369-7412 (ISSN-L)
  • 1467-9868
DOI
10.1111/j.2517-6161.1978.tb01663.x
OpenAlex
W131298435
Cited by
224
References
0
FWCI
2.48
Type
journal article
Language
en
Open access
closed
Licence
academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model (vor)

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Gabriel, K. R. (1978). Least Squares Approximation of Matrices by Additive and Multiplicative Models. Journal of the Royal Statistical Society Series B: Statistical Methodology. https://doi.org/10.1111/j.2517-6161.1978.tb01663.x

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