Generalized additive models(GAMs) provide a general framework for extending a standard linear model by allowing non-linear functions of each of the variables, while maintaining additivity. GAMs can be applied with both quantitative and qualitative responses.

GAMs for Regression Problems

$$y_i=\beta_0+f_1(x_{i1})+f_2(x_{i2})+\ldots+f_p(x_{ip})+\epsilon_i$$

It is called an additive model because we calculate a separate $f_j$ for each $X_j$, and then add together all of their contributions.

Advantages and limitations of a GAM:

• GAMs allow us to fit a non-linear $f_j$ for each $X_j$, so that we can automatically model non-linear relationships that standard linear regression will miss. This means that we don't need to manually try out many different transformations on each variable individually.
• The non-linear fits can potentially make more accurate predictions for the response $Y$.
• Because the model is additive, we can still examine the effect of each $X_j$ on $Y$ individually while holding all of the other variables fixed. Hence if we are interested in inference, GAMs provides a useful representation.
• The smoothness of the function $f_j$ for the variable $X_j$ can be summarized via degree of freedom.
• The main limitation of GAMs is that the model is restricted to be additive. With many variables, important interactions can be missed.

GAMs provide a useful compromise between linear and fully nonparametric models.