Consider a GAM model, expressed in
mgcv just to fix ideas:
my_model <- gam(y ~ ti(x1)+ti(x2) + ti(x1, x2), method= "REML")
The model is linear in the parameters, right? Each smooth is a linear combination of basis functions, which are independent of the data set (unless I use
bs = "ad"). Thus the model is linear in the parameters, which are the coefficients of the basis functions. Right? And this should be true, whether or not there are interaction terms - it doesn't really matter. The only exception are adaptive smooths, because in that case the coefficients of the basis functions are themselves functions of the covariates (
x2, in my example). Correct?