ABSTRACT
A generalized additive model (Hastie and Tibshirani, 1986, 1990) is a generalized
linear model with a linear predictor involving a sum of smooth functions of covari-
ates. In general the model has a structure something like
g(µi) = X ∗ iθ + f1(x1i) + f2(x2i) + f3(x3i, x4i) + . . . (3.1)
where
µi ≡ E(Yi) and Yi ∼ some exponential family distribution. Yi is a response variable,X
∗ i is a row of the model matrix for any strictly parametric
model components, θ is the corresponding parameter vector, and the fj are smooth functions of the covariates, xk. The model allows for rather flexible specification of the dependence of the response on the covariates, but by specifying the model
only in terms of ‘smooth functions’, rather than detailed parametric relationships, it
is possible to avoid the sort of cumbersome and unwieldy models seen in section
2.3.4, for example. This flexibility and convenience comes at the cost of two new
theoretical problems. It is necessary both to represent the smooth functions in some
way and to choose how smooth they should be.
