Reference for the idea that a simpler model can be used when the range of data values is smaller

Question

When we build a statistical-physical model, generally, a simpler model can be justified when the range of data-values is smaller.

I can't be the first person to use this idea, but I also can't find it mentioned anywhere.

As a simple example of what I am talking about, consider a falling rock. If the rock only falls one metre, then a constant constant-acceleration model is probably useful. If the rock falls a kilometre, then I likely need to consider aerodynamics. If the rock falls 1000 kilometres, orbital-mechanics will need to be there.

My actual situation doesn't involve falling rocks, but I do suggest limiting the range of data-values as a means of managing the required complexity of a model.

Can anybody recall a name for the concept - or suggest a reference?

Answer 1

I think that this is not a reference-able concept, it's just about relative error. What you call "range of data values" is usually called "scale", and you would just say that a certain theory is enough descriptive for this scale. In the example of the rock, $F=-mg$ is enough to describe the dynamics of the rock, in the sense that the error you make by the theory is smaller than the typical measurement error. In this particular case, is called linearisation, and is typical when you have non lineal equations of motion and you want to use them in a small neighbourhood.

In statistics, I would see an equivalent in error measures that contain a cost for the amount of parameters of the model (Akaike or Bayesian Information Criteria, for example). You measure the error you make for models with a different amount of parameters, and at some point the error reduction that you make for adding a new parameter would not be enough to compensate the cost you predefined; be careful with those though, since the definition of the cost is crucial.