I have some data which consists of an exponential process in time convolved with a gating function. I am fitting the underlying exponential using the least squares method. (I also have experimental data for the gating function itself that I convolve with perfect exponentials to get the "fit" and find best parameters to minimize difference between fit and data.)
If I assume the exponential is monoexponential, eg $d(t)=Ae^{-t/\tau}+C$, I get a pretty good fit. It's also consistent between the two datasets I have (which are simultaneously gathered, basically the two sides of the gating) to two decimal places, so let's say it's 4.08ns. If I assume the exponential has two components, eg $d(t)=A_1e^{-t/\tau_1}+A_2e^{-t/\tau_2}+C$, I get a basically perfect fitting where $\tau_1$ and $\tau_2$ are similar to two decimal places but quite different than for monoexponential fitting, say 4.32ns (the sum of squared residuals is an order of magnitude lower).
Basically, how can it be that a monoexponential fit gives a significantly different exponential parameter than a biexponential fit which indicates a monoexponential in that it gives essentially a single exponential parameter? Am I overfitting in the second case? How can I know if I'm overfitting?
I plotted the monoexponential fit with 4.3ns on top of the data and the monoexponential fit with 4.0ns. To the eye 4.3ns isn't bad / is just a little bit further from the data and crosses over it in places where 4.0ns overlaps with it - you can tell 4.0ns is a better monoexponential fit. Is the physical reality somehow that it's biexponential with two exponents that are very similar? Why is this so unintuitive to me?
Also it's clear that the meaning of the sum of squared residuals in terms of how good the fit is is highly dependent on what the general magnitudes of the datapoints are. My data involves normalization, so e.g. the two datasets sum to 1 at every timepoint, but even with the varying contrast in different datasets it's clear I can't compare those - only within one dataset. Is there a guideline for what a "good" fit is as a ratio to the general magnitude/contrast? How can I interpret the squared sum of residuals more intelligently, or is there another metric to understand "goodness of fit" and overfitting?
