  In the present paper, we prove a new theorem, resulting in an update formula
for linear regression model residuals calculating the exact k-fold
cross-validation residuals for any choice of cross-validation strategy without
model refitting. The required matrix inversions are limited by the
cross-validation segment sizes and can be executed with high efficiency in
parallel. The well-known formula for leave-one-out cross-validation follows as
a special case of the theorem. In situations where the cross-validation
segments consist of small groups of repeated measurements, we suggest a
heuristic strategy for fast serial approximations of the cross-validated
residuals and associated Predicted Residual Sum of Squares (PRESS) statistic.
We also suggest strategies for efficient estimation of the minimum PRESS value
and full PRESS function over a selected interval of regularisation values. The
computational effectiveness of the parameter selection for Ridge- and Tikhonov
regression modelling resulting from our theoretical findings and heuristic
arguments is demonstrated in several applications with real and highly
multivariate datasets.
