If you are considering the information theoretical point of view, given x3
and x4
then x3/x4
adds no information.
However, before we rush into conclusions, one must recall that there are more aspects.
The first one is the ability to represent the concept. Let's consider as case in which your features are x3
and x4
and the concept is x3/x4
.
The only problem is that I force you to use a decision tree as the model.
How can you represent that?
Note that even if I allow you to choose any tree, the representation will be bad.
The second problem is that the concept is not given in advance and therefor the algorithm should search for it. Most supervised learning algorithms are actually looking for correlations between the concept and the features. A relation like 1/x
usually cause them problem since the behavior is not linear in x.
So, in case that you have ideas for features that don't add information but ease representation or search, adding them might help.
You are right asking where should I stop? It might be that there is a different combination of features that will lead to a better model.
Unfortunately, we don't have a good solution for that.
Consider the problem of building good features, let's say features that reduce the size of the model. We can restrict our self to features that can be represented by trees. However, building a small tree for a data set is an NP-complete problem, which means that we don't have an efficient algorithm for that.
More than that, you can claim that given a feature building problem you can reduce classification into it (e.g., build a feature that represents the concept). Therefore, feature building is as hard as classification.