This may be a simple problem, but I want to be thorough in setting up my problem as I'd like to know why I should proceed in one of two ways (or another if someone thinks it is suitable), so please bare with me.
I'm trying to calibrate the position of an array of 18 avalanche photodiode (APD) detectors. In the experiment, the signal is approximated to come along a single x-axis. To calibrate this, I have an LED that I can move along the axis into my experiment and I can see the intensity increase in a single APD channel as it goes past.
I can only get my LED to go past channel 3-12 of my array, but based on previous experiments, where the experimental setup isn't expected to have changed too much, I expect the signal each APD detects to be evenly spaced.
I have a set of x coordinates that I was able to measure, each with an associated error with position measurements when setting up the calibration, as well as a finite width of the signal detected in a single APD channel.
I use these errors to fit a weighted linear function to the data using scipy.optimize.curve_fit. So as a function of channel number, $n$, I can calculate the position, $x$, from the simple formula
$x = a \times n + b$
where $a$ and $b$ are the coefficients given out from the fitting function. I can also get out the co-variance matrix, and I can also calculate the mean square error (MSE) of the fit to my data points.
Question: how should I estimate the error of the position of the channels?
1) Combine the error of the two fit parameters $a$ and $b$ in quadrature, given as the diagonals of the co-variance matrix.
2) Alternatively, quoting the MSE as the error on any reading.
My error will be larger through method 1, and it seems more robust, especially if I want to use the fit to extrapolate the channels I wasn't able to reach with the LED. My instinct is to add the error of the coefficients in quadrature, but if that's so, what is the MSE actually good for? Or is it just a non-normalised number that gives me an indication of the goodness of the fit to the range I have?