Orthogonalization of Vectors with Minimal Adjustment

A new transformation is proposed that gives 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. If the vectors have been normalized, then the transformation minimizes the sum of the squared correlations between each vector and its associated component. The transformation has a strong dilution property: duplicating a vector, perhaps several times, has no effect on the orthogonal components that correspond to non-duplicated vectors. Properties of the transformation give it varied uses, notably as a diagnostic tool for identifying collinearities and as a method of choosing Bayesian prior weights for model averaging. These applications are described and we also discuss regression and experimental design, broader contexts in which the transformation should prove useful.