Linear combine non-linear transformations Could someone tell me what it is called if you linear combine a non-linear transformation such as:
$$y_i = \beta_1 f(x_{1i}) + \beta_2 f(x_{2i}) + \ldots + \beta_n f(x_{ni}),$$
where $f(\cdot)$ is a non-linear transformation relating the predictor to the response variable? To me this seems similar to a Generalized Additive Model, but instead of having the "link", the "smoothed" expressions are provided.
 A: If you specify the $f$ (or $f$s), this is called feature engineering, and then you are doing linear regression on nonlinear basis functions.
When you present the data to a linear regression, it does not know or care how you got the features. If they are raw measurements, they're just numbers to be thrown into the calculation. If you were thoughtful about how to use features and decided that you should be squaring a feature, those squared values are just numbers to be thrown into the calculation. Your interpretation of the regression will depend on what you did, but your $f(x_i)$ features just as easily could be defined as $z_i$ for a regression equation of:
$$\hat y_i = \hat \beta_1 z_{1i} + \hat \beta_2 z_{2i} + \ldots + \hat \beta_n z_{ni}$$
Jeffrey Miller, known as MathematicalMonk on YouTube, has a good video about this. You don't have to use the same $f$ for each $x$ variable, and you can have basis functions of multiple variables (such as interactions) in your linear combination.
