Currently I'm fitting 2D points with a degree-2 polynomial, using least squares:
$$ b = (X^T X)^{-1} X^T y $$
Example result:
My problem is that I would need all parabolas to have a maximum. Meaning: if the resulting polynomial is:
$$ a x^2 + b x + c $$
then a < 0.
Any hints for solving this?
If you are doing a unconstrained least-squares fit, you don't have much control. In your case, it looks like your data is concentrated in a relatively small region, and it looks like it tends to be increasing.
Why does the fit need to have a < 0?
It would help if your plot was zoomed in so the bulk of the data filled the image.