Pseudosplines

SUMMARY We describe a method for constructing a family of low rank, penalized scatterplot smoothers. These pseudosplines have shrinking behaviour that is similar to that of smoothing splines. They require two ingredients: A basis and a penalty sequence. The smoother is then computed by a generalized ridge regression. The family can be used to approximate existing high rank smoothers in terms of their dominant eigenvectors. Our motivating example uses linear combinations of orthogonal polynomials to approximate smoothing splines, where the linear combination and the penalty sequence depend on the particular instance of the smoother being approximated. As a leading application, we demonstrate the use of these pseudosplines in additive model computations. Additive models are typically fitted by an iterative smoothing algorithm, and any features other than the fit itself are difficult to compute. These include standard error curves, degrees of freedom, generalized cross-validation and influence diagnostics. By using a low rank pseudospline approximation for each of the smoothers involved, the entire additive fit can be approximated by a corresponding low rank approximation. This can be computed exactly and efficiently, and opens the door to a variety of computations that were not feasible before.