ABSTRACT

A generalized additive model is a generalized linear model with a linear predictor involving a sum of smooth functions of covariates. This chapter illustrates how generalized additive models (GAMs) can be represented using basis expansions for each smooth, each with an associated penalty controlling function smoothness. Estimation can be carried out by penalized regression methods, and the appropriate degree of smoothness for the fj can be estimated from data using cross validation or marginal likelihood maximization. To avoid obscuring the basic simplicity of the approach with a mass of technical detail, the most complicated model considered will be a simple GAM with two univariate smooth components. Taylor's theorem implies that polynomial bases will be useful for situations in which interest focuses on properties of f in the vicinity of a single specified point. It clearly makes sense to use bases that are good at approximating known functions in order to represent unknown functions.