I have some data, and I expect that part of the data will fit to a power law, and part of the data to a line. Here's a made up example of some similar data:
Where I've made the red part a quadratic with some error, and the blue part a line with some error, just for illustration.
Now supposed I don't know where this clear break between the functions is because it was data I just took in. I want to essentially find this point by choosing points throughout the data range, and for each point, suppose that's the "break point", and try fitting a power law to the left of it and a line to the right, and getting the R Squared value for each section, and minimizing the total error.
My question is, if I have the R Squared for each section, what's the best way to minimize the "total error"? Do I want to minimize simply $R_{power}^2 + R_{linear}^2$?
Edit: To clarify, I don't know the actual exponent of the power law, or the slope of the line. I'll be fitting a general power law of the form $ax^b$ and a line $mx+c$ to the two regions (so it's not a matter of just finding the divider between the region for two known functions).