# Propagation of error in fitting two sets of data to each other

I have two sets of experimental data: $\phi(t)$ and $I(t)$. In theory they are related to each other as: $\phi(t) = nI(t)$. By fitting these curves together I can find the value of $n$ (which is a constant). I can calculate the error of this fit using residue and covariance.

Now, for each point $t$ of my data sets I also have error of $\phi(t)$ and $I(t)$. How to propagate this error through the fitting procedure, so the final error of the $n$ parameter is also influenced by them?

• What assumptions are you willing/able to make about the measurements of $\phi(t_{i})$ and $I(t_{i})$? Are they indendent? Normally distributed? With known standard deviations? Oct 29, 2017 at 21:52
• $I(t)$ is intensity of a femtosecond pulse, $phi(t)$ is its phase. I am measuring the process of self-phase modulation that imprints phase on the pulse that is proportional to intensity. Both those values are calculated through generalized projections algorithm from sum-frequency generation in a method called FROG (frequency-resolved optical gating). Shape of those functions is similar to gaussian, I can provide exemplary data set. Error of those functions is calculated using FROG-bootstrap method, which gets me standard deviation for each point $t_i$. Oct 30, 2017 at 7:13