• Basis penalty smooth;
  • Differential equation smoothing;
  • Finite window smoothing;
  • Known boundary smoothing;
  • Spline

Summary.  Conventional smoothing methods sometimes perform badly when used to smooth data over complex domains, by smoothing inappropriately across boundary features, such as peninsulas. Solutions to this smoothing problem tend to be computationally complex, and not to provide model smooth functions which are appropriate for incorporating as components of other models, such as generalized additive models or mixed additive models. We propose a class of smoothers that are appropriate for smoothing over difficult regions of inline image2 which can be represented in terms of a low rank basis and one or two quadratic penalties. The key features of these smoothers are that they do not ‘smooth across’ boundary features, that their representation in terms of a basis and penalties allows straightforward incorporation as components of generalized additive models, mixed models and other non-standard models, that smoothness selection for these model components is straightforward to accomplish in a computationally efficient manner via generalized cross-validation, Akaike's information criterion or restricted maximum likelihood, for example, and that their low rank means that their use is computationally efficient.