We’re often modeling non-monotonic functions. For example, performance at just about any task increases with age (babies can’t do much!) and then eventually decreases (dead people can’t do much either!). Here’s an example from a few years ago:
A function g(x) that increases and then decreases can be modeled by a quadratic, or some more general functional form that allows different curvatures on the left and right sides of the peak, or some constrained nonparametric family.
Here’s another approach: an additive model, of the form g(x) = g1(x) + g2(x), where g1 is strictly increasing and g2 is strictly decreasing. This sort of model gets us away from the restrictions of the quadratic family—it’s trivial for g1 and g2 to have different curvatures—but also it is a conceptual step forward in that it implies two different models, one for the process that causes the increase and one for the process that causes the decrease. This makes senses, in that typically the increasing and decreasing processes are completely different.
This is an example of the Pinocchio principle.
P.S. The original title of this post was “Additive models for non-monotonic functions,” but that seemed like the most boring thing possible, so I decided to go clickbait.