ABSTRACT
This chapter discusses spline smoothers, and the equivalence between quadratically penalized smoothers and Gaussian random effects/fields. It considers one dimensional spline smoothing, and then computationally efficient reduced rank penalized regression splines. The chapter examines several one dimensional penalized regression smoothers, including adaptive smoothers. It describes constraints, the 'natural' parameterization of a smooth and effective degrees of freedom, followed by multidimensional smoothers. First isotropic smoothers: thin-plate and other Duchon splines, splines on the sphere and soap film smoothers. Isotropy is usually inappropriate when the arguments of a smooth have different units. Scale invariant 'tensor product' smoothers are therefore constructed next, and consideration given to the notion of a smooth interaction, and to decompositions involving smooth main effects and smooth interactions. The chapter explains the duality between smooths and Gaussian random effects/fields and of Gaussian Markov random field smoothers and Gaussian process smoothers.
