Thin plate regression splines

I discuss the production of low rank smoothers for d ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thinplate spline smoothing problem, and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin-plate spline smoothing problem given the dimension of the basis used to construct the smoother. By making use of Lanczos iteration the basis change and truncation is computationally efficient. The smoothers allow the use of approximate thin-plate spline models with large datasets; avoid the problems associated with “knot placement” that usually complicate modelling with regression splines or penalized regression splines; provide a sensible way of modelling interaction terms in GAMs; provide low rank approximations to generalized smoothing spline models, appropriate for use with large datasets; provide a means for incorporating smooth functions of more than one variable into non-linear models and improve the computational efficiency of penalized likelihood models incorporating thin-plate splines. Given that the approach produces spline like models with a sparse basis, it also provides a natural way of incorporating un-penalized spline like terms in linear and generalized linear models, and these can be treated just like any other model terms from the point of view of model selection, inference and diagnostics.