I'm trying to fit an exponential with noise (which in this case is a constant $c$) like this one
$$y(x)=αe^{−αx}+c,$$
having $(x_i, y_i)$ values (So $α$ and $c$ are unknown and are the ones that I want to find out). Without noise I simply linearize the values and then I apply the Least Square Method, but with noise I have no idea how to do. Are there any formula to do it? Thank you.
For two data points, $y = ae^{-ax}+c$ and $w = a e^{-av}+c$.
Subtracting, $y-w =a(e^{-ax}-e^{-av}) $. If $v$ is a lot larger that $x$, then $y-w \approx ae^{-ax} $. At this point, the Lambert-W function could be used to determine $a$. Then $c$ could be gotten.
This would give a set of $a$ and $c$ values which could be averaged or looked at to see how much they vary.