Downdating the Moore-Penrose Generalized Inverse for Cross-Validation of Centred Least Squares Prediction

Abstract

SUMMARY Given a p × p symmetric matrix B and a p-vector c outside the range of B, a simple expression is formulated for the action of the Moore-Penrose generalized inverse B + on a vector b in the range of B, in terms of b, c and (B + cc‱)+. A special case of this formula gives a ‘downdating’ formula for the vector of least squares regression coefficients based on the Moore-Penrose inverse of the centred sum of squares and products matrix. The formula is analogous to that given by A. Albert for uncentred least squares fitting. An explicit downdating formula for the Moore-Penrose inverse B + itself is derived, and then more general downdating expressions for the inverse and the vector B + b when the p-vector c is replaced by a matrix array C of such vectors outside the range of B.