The minimal residual QR-factorization algorithm for reliably solving subset regression problems

A new algorithm to solve test subset regression problems is described, called the minimal residual QR factorization algorithm (MRQR). This scheme performs a QR factorization with a new column pivoting strategy. Basically, this strategy is based on the change in the residual of the least squares problem. Furthermore, it is demonstrated that this basic scheme might be extended in a numerically efficient way to combine the advantages of existing numerical procedures, such as the singular value decomposition, with those of more classical statistical procedures, such as stepwise regression. This extension is presented as an advisory expert system that guides the user in solving the subset regression problem. The advantages of the new procedure are highlighted by a numerical example.