Orthogonalization of vectors with minimal adjustment

Paul H. Garthwaite, Frank Critchley, Karim Anaya-Izquierdo, Emmanuel Mubwandarikwa

Research output: Contribution to journalArticle

7 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)787-798
Number of pages12
JournalBiometrika
Volume99
Issue number4
Early online date18 Sep 2012
DOIs
Publication statusPublished - Dec 2012

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Orthogonalization
Adjustment
new variety
statistical analysis
Diagonal of a matrix
Bayesian Model Averaging
Weights and Measures
Collinearity
degradation
Positive definite matrix
Correlation Matrix
Global Optimum
One to one correspondence
Orthogonality
Statistical method
Statistical methods
Upper bound
Decomposition
Converge
Decompose

Cite this

Garthwaite, P. H., Critchley, F., Anaya-Izquierdo, K., & Mubwandarikwa, E. (2012). Orthogonalization of vectors with minimal adjustment. Biometrika, 99(4), 787-798. https://doi.org/10.1093/biomet/ass041

Orthogonalization of vectors with minimal adjustment. / Garthwaite, Paul H.; Critchley, Frank; Anaya-Izquierdo, Karim; Mubwandarikwa, Emmanuel.

In: Biometrika, Vol. 99, No. 4, 12.2012, p. 787-798.

Research output: Contribution to journalArticle

Garthwaite, PH, Critchley, F, Anaya-Izquierdo, K & Mubwandarikwa, E 2012, 'Orthogonalization of vectors with minimal adjustment', Biometrika, vol. 99, no. 4, pp. 787-798. https://doi.org/10.1093/biomet/ass041
Garthwaite, Paul H. ; Critchley, Frank ; Anaya-Izquierdo, Karim ; Mubwandarikwa, Emmanuel. / Orthogonalization of vectors with minimal adjustment. In: Biometrika. 2012 ; Vol. 99, No. 4. pp. 787-798.
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