 Solution

A Gaussian generalised additive model with the identity link specifiesThe $f_j$ are unknown smooth functions estimated from the data, rather than necessarily straight lines. The model is additive in their effects, so it does not automatically contain interactions between covariates. To identify the intercept separately from the smooths, impose a constraint such as ${\sum }_i{f}_j(x_{ij})=0$ for each $j$. Linear terms can also be included alongside smooth terms. More generally a specified link function can make $g(E(Y_i))$ additive; for the normal identity-link model the mean itself has the displayed form. Smoothness is typically controlled through a basis and a roughness penalty, preventing an unrestricted smooth from interpolating all the noise.