# Data fitting with repeated measurements

we have an experiment to excite a system with some energy, then measure the decay as a function of time. We are measuring data at 4 times $t$ to fit to an exponential model: $y = a \exp(-t/p) + c$, where we fit the parameters $a, p$, and $c$. I should probably mention that there is not one excitation and 4 measurements, but rather we excite 4 times, and adjust our time-from-excitation to measure -- so each measurement is taken independently.

We have pretty good signal-to-noise with respect to the measurement device, however the object under consideration may or may not move during the experiment resulting in some unpredictable error in one or more data points. Parameter $p$ is the most important, and we know a physical range a-priori for this parameter.

We must make this measurement in a fixed amount of time and we have time to collect a maximum of 8 measurements. Right now, we are collecting the data twice at each of the 4 time points, so we have basically two distinct experiments.

I have a lot of questions, actually :), but one I find interesting is whether it is better to collect two measurements at 4 time points, or if it is better to collect 8 measurements at different time points (supposedly to better characterize the decay curve). We've done the experiment at up to 16 time points and found good agreement with the 4 time point experiment. Other questions I have pertain to how to detect bad measurements in the 2X 4 time point data sets.

Any ideas would be greatly appreciated! I am not too well-learned in statistics, so if you have some suggestions on some background reading or material pertinent to this problem, that would be awesome!

Thanks!!