# 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? – Brian Borchers Oct 29 '17 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 $t_i$. – KabaT Oct 30 '17 at 7:13

## 1 Answer

This is an example of what is called an errors in variables regression model. Rather than having an independent variable that is known exactly and a dependent variable that is measured with noise, but the independent and dependent variables are measured with noise.

The most commonly used technique for fitting linear errors in variables model is the method of total least squares. See for example the Wikipedia article on TLS.