I have been working with a set of data which is set in an engineering discipline. However my aim is predictive in nature, i.e., I need to get the relationship between the parameters as well as predict the variable of interest using its dependent features.
I am aware of machine learning techniques like ANN, random forests, ensemble techniques with which I can predict the given value.
$$Y = a_1X_1 + a_2X_2 +...$$
However I started out with a simple regression (multiple regression) $(n\gg p)$ and it gave me a not too high residual error. I am certain that these errors can be further minimized using more complicated techniques, and this is because many of the features $(X_1, X_2, \ldots)$ share a non-linear relationship with the variable $(Y)$ to be predicted. However is there any way using regression I can prove / see whether any of the features are related to the predictor linearly?
Another concept which bothers me is that if I was to change my regression to include non linear feature terms as shown below, would that technically still be a linear regression or would the model (regression model) start to capture some of the non linear effects of the model
$$Y = a_1X_1 + a_2X_1^2 + a_3X_2 + a_4X_2^2 + \ldots$$