I'm using Gaussian Process regression for the first time to model the unknown energy efficiency of a compressor which I know is a smooth, non-linear relationship that looks something like the line in this plot:
One option (option 1a) is to have the GP model map load (kW) to power (kW) just like in the figure above. However, in this case I would want to specify a prior for the mean function, at least some ascending line since I know power consumption is not constant and increases with load.
For simplicity, let's say I have chosen power = 0.7 x load as the prior.
As mentioned in this post, the prior of the GP is usually set to zero in most text book examples, and is instead subtracted from the data before fitting the GP to it (option 1b).
Q1: Are these two options exactly equivalent or are there some effects I am not aware of?
However, a better option (option 2) I think, would be to re-cast the problem and have the GP model map load to specific energy, where specific energy is defined as power / load (kW/kW). This is a different relationship and looks something like this:
In this case, the prior would simply be a constant value, say 0.7.
The person who wrote this answer argued that this is a better option because "the zero mean (or, for what's worth, also the constant mean) GP kind of sucks at prediction far away from the training data" which is true. And a zero prior is not an option here because I do want to do extrapolation.
Q2: What are the differences (if any), as well as pros and cons of this second option compared to options 1a and 1b?
Doesn't option 2 also change the way measurement noise (error) is modelled? (In this application, I think the measurement noise is probably additive on the power measurements and its variance is most likely independent of the load).
Maybe there are other implications of this option too that I should be aware of.
