With two arrays $x$ and $y$ giving exponential decay as given in this question, an exponential function in the form $y = Ae^{-Bt} + C$ is fitted in this answer. I want to generalize to the fitting function in the form
$$y = 1 - (A_1(1 - e^{-B_1t}) + A_2(1 - e^{-B_2t}) + A_3(1 - e^{-B_3t})+\dots);$$
basically using $2, 3, ..., N$ exponential terms. Any comments on how to do it?
My data is slightly different from a pure exponential decay (as described in the above links) and a single exponential term is not sufficient to fit the data properly.
The principle of a method which can be generalised is given in https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales
Case of one exponential : Pages 16-17.
Case of two exponentials : Page 72.
The case of three exponentials is not explicitely treated in the paper. In addition see below :
On the same way one could treat the cases of more exponentials. But this would probably be of no practical use because the deviations due to the scatter of the experimental data. Even with three exponentials the numerical calculation becomes doudtful. Too many parameters to optimise in the model supposes to have big data with low scatter.
For more robustness, better compare to a model with two exponential instead of three. See below :