FOURIER SMOOTHER AND ADDITIVE-MODELS

Suppose the observations (t(i), y(i)), i = 1,...,n, follow the model E(y\t) = g1(t1) +...+ g(r)(t(r)), where g(j) are unknown functions. The estimation of the additive components can be done by approximating g(j) with a function made up of the sum of a linear fit and a truncated Fourier series of cosines and minimizing a penalized least-squares loss function over the coefficients. This finite-dimensional basis approximation, when fitting an additive model with r predictors, has the advantage of reducing the computations drastically, since it does not require the use of the backfitting algorithm. The cross-validation (CV) [or generalized cross-validation (GCV)] for the additive fit is calculated in a further O(n) operations. A search path in the r-dimensional space of degrees of freedom is proposed along which the CV (GCV) continuously decreases. The path ends when an increase in the degrees of freedom of any of the predictors yields an increase in CV (GCV). This procedure is illustrated on a meteorological data set.