Orthogonalization of vectors with minimal adjustment

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

Biometrika, 2012, vol. 99, issue 4, 787-798

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.

Date: 2012
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