Given the multiple linear regression model:

$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \dots + \beta_p x_{ip} + \epsilon_i$

A natural way to extend the multiple linear regression model is to replace each linear component $\beta_j x_{ij}$ with a smooth nonlinear function $f_j(x_{ij})$. We would then write the model as:

$y_i = \beta_0 + \sum_{j=1}^p f_j(x_{ij}) + \epsilon_i$