  It is shown how to efficiently and accurately compute and optimize a range of
cross validation criteria for a wide range of models estimated by minimizing a
quadratically penalized smooth loss. Example models include generalized
additive models for location scale and shape and smooth additive quantile
regression. Example losses include negative log likelihoods and smooth quantile
losses. Example cross validation criteria include leave-out-neighbourhood cross
validation for dealing with un-modelled short range autocorrelation as well as
the more familiar leave-one-out cross validation. For a $p$ coefficient model
of $n$ data, estimable at $O(np^2)$ computational cost, the general $O(n^2p^2)$
cost of ordinary cross validation is reduced to $O(np^2)$, computing the cross
validation criterion to $O(p^3n^{-2})$ accuracy. This is achieved by directly
approximating the model coefficient estimates under data subset omission, via
efficiently computed single step Newton updates of the full data coefficient
estimates. Optimization of the resulting cross validation criterion, with
respect to multiple smoothing/precision parameters, can be achieved efficiently
using quasi-Newton optimization, adapted to deal with the indefiniteness that
occurs when the optimal value for a smoothing parameter tends to infinity. The
link between cross validation and the jackknife can be exploited to achieve
reasonably well calibrated uncertainty quantification for the model
coefficients in non standard settings such as leaving-out-neighbourhoods under
residual autocorrelation or quantile regression. Several practical examples are
provided, focussing particularly on dealing with un-modelled auto-correlation.
