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In multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory, if not incorrect, if the prediction vectors are not orthogonal. Proposed is an estimation procedure based on adding small positive quantities to the diagonal of X′X. Introduced is the ridge trace, a method for showing in two dimensions the effects of nonorthogonality. It is then shown how to augment X′X to obtain biased estimates with smaller mean square error.
ridge regression is equivalent to the following linear model Y = mu + X betas + E with X being NxP and E ~ MVN(0,I_N tau^-1) where each beta is assumed to have prior N(0,sigma_b^2/(P tau)), which is also equivalent to the following linear mixed model Y = X betas + Z u + E where u is a random effect distributed as MVN(0, lambda tau^-1 K) with the K matrix being (1/P)XX^T (ie. N x N)
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