PSEUDOSPLINES

We describe a method for constructing a family of low rank, penalized scatterplot smoothers. These pseudosplines have shrinking behaviour th at is similar to that of smoothing splines. They require two ingredien ts: a basis and a penalty sequence. The smoother is then computed by a generalized ridge regression. The family can be used to approximate e xisting high rank smoothers in terms of their dominant eigenvectors. O ur motivating example uses linear combinations of orthogonal polynomia ls 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 fea tures other than the fit itself are difficult to compute. These includ e standard error curves, degrees of freedom, generalized cross validat ion and influence diagnostics. By using a low rank pseudospline approx imation for each of the smoothers involved, the entire additive fit ca n be approximated by a corresponding low rank approximation. This can be computed exactly and efficiently, and opens the door to a variety o f computations that were not feasible before.