On Neighbourhood Cross Validation

Many varieties of cross validation would be statistically appealing for the estimation of smoothing and other penalized regression hyperparameters, were it not for the high cost of evaluating such criteria. Here it is shown how to efficiently and accurately compute and optimize a broad variety of cross validation criteria for a wide range of models estimated by minimizing a quadratically penalized loss. The leading order computational cost of hyperparameter estimation is made comparable to the cost of a single model fit given hyperparameters. In many cases this represents an $O(n)$ computational saving when modelling $n$ data. This development makes if feasible, for the first time, to use leave-out-neighbourhood cross validation to deal with the wide spread problem of un-modelled short range autocorrelation which otherwise leads to underestimation of smoothing parameters. It is also shown how to accurately quantifying uncertainty in this case, despite the un-modelled autocorrelation. Practical examples are provided including smooth quantile regression, generalized additive models for location scale and shape, and focussing particularly on dealing with un-modelled autocorrelation.