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Orthogonalization of vectors with minimal adjustment

Author

Listed:
  • Paul H. Garthwaite
  • Frank Critchley
  • Karim Anaya-Izquierdo
  • Emmanuel Mubwandarikwa

Abstract

Two transformations are proposed that give orthogonal components with a one-to-one correspondence between the original vectors and the components. The aim is that each component should be close to the vector with which it is paired, orthogonality imposing a constraint. The transformations lead to a variety of new statistical methods, including a unified approach to the identification and diagnosis of collinearities, a method of setting prior weights for Bayesian model averaging, and a means of calculating an upper bound for a multivariate Chebychev inequality. One transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors, and is determined using a new algorithm that also provides the decomposition of a positive-definite matrix in terms of a diagonal matrix and a correlation matrix. The algorithm is shown to converge to a global optimum. Copyright 2012, Oxford University Press.

Suggested Citation

  • Paul H. Garthwaite & Frank Critchley & Karim Anaya-Izquierdo & Emmanuel Mubwandarikwa, 2012. "Orthogonalization of vectors with minimal adjustment," Biometrika, Biometrika Trust, vol. 99(4), pages 787-798.
    • Handle: RePEc:oup:biomet:v:99:y:2012:i:4:p:787-798
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    File URL: http://hdl.handle.net/10.1093/biomet/ass041
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