I have a linear model with one continuous dependent variable ($y$) and one explanatory variable ($x$). The model: $y = b_0 + b_1 * x$; now I and need to add a categorical feature to it. The problem is that training for each category should result in its own curve ($b_0, b_1$ pair). Each category is a different process. So I cannot just use a linear regression and add just add $b_2*x_2$.
I'm tempted to use the simplest way: train a separate model for each process. But my feeling is this might fall in one of well-known problems as a single model. The actual problem is in ML field.