Generalized Additive Models › Advantages

The additive restriction limits the number of functions that need estimating. For example, a Cartesian grid with $m$ positions per predictor contains $m^p$ grid points, while $p$ separate one-dimensional grids contain $pm$ positions. A GAM imposes that lower-dimensional additive structure rather than estimating an unrestricted surface. The comparison describes the representation, not a universal sample-size bound. The additive restriction represents a sum of $p$ one-dimensional functions instead of an unrestricted surface. Their coefficients still have to be fitted jointly, and computation grows with the total basis dimension.

Second, a GAM remains interpretable. Because the components enter additively, we can plot each $f_j$ as a curve and read off how the predicted value (or the transformed mean) changes as a function of $x_j$, holding the others fixed. These are direct plots of model terms on the additive-predictor scale. With a nonlinear link, they differ from response-scale partial-dependence curves, which average predictions over other inputs; see Part XI: Model Interpretation. That makes GAMs attractive in domains where the answer "the model predicts this" is not enough and stakeholders want to know why.

Third, a GAM extends the machinery of generalized linear models to non-Gaussian responses without abandoning interpretability. We can use the same spline bases we built for continuous responses inside a logistic, Poisson, or gamma regression by adding a link function. This gives us nonlinear logistic and count regression through penalized likelihood. As a consequence, when a linear GLM is clearly misspecified, such as a binary outcome whose logit is obviously curved in a predictor, a GAM is a natural upgrade.

Finally, the smoothing penalty plays the role of a continuous regularizer. Rather than choosing the number of knots arbitrarily, we let the data guide how wiggly each curve should be via cross-validation or marginal likelihood. This is the same idea ridge regularization will formalize in the next chapter Ridge Regularization (L2 Regularization); in fact, a GAM with a difference penalty is a generalized quadratic-penalty regression on an expanded feature space; the penalty matrix differs from ordinary ridge.