Propagate error through `FindFit`

Question

I have several sampled points $(x_i, y_i)$ with mutually independent error on the y coordinate $\sigma_{y_i}$. For simplicity, I'm assuming no error in the x coordinates ($\sigma_{x_i} \ll \sigma_{y_i}$)

I'm attempting to find a model this set of points, and propagate the error (along with any appropriate error from the fit, if applicable) to the model. Is there a simple way to do this? I'm completely unaware about how to normally propagate errors in Mathematica, so I've been doing it by hand using the standard formula for independent errors (where $f(x_1, x_2,... x_i)$):

$$\sigma_f = \sqrt{\sum_{i}{\left(\frac{\partial f}{\partial x_i} \sigma_{x_i}\right)}^2}$$

And while I could re-implement FindFit using the mathematical definition of a least squares fit, (ie, take the gradient of the residuals and solve the system,) then figure out how to throw that into the above formula, I think surely there must be a better way.

If it helps, specifically, the model I'm attempting to fit is $A \cos{\pi x / a}$ (with a constraint on $a$ to avoid a Nyquist issue, of course.) I'd like to obtain $a$, $A$, $\sigma_a$, and $\sigma_A$.

Answer 1

Using NonlinearModelFit results in a model object from which some relevant data can be extracted. See:

fit = NonlinearModelFit[data, {A Cos[\[Pi] x/a], a > 0}, {A, a}, x, Method -> NMinimize];
fit["Properties"]
fit["MeanPredictionBands"]

For a list of properties and an example of one of them. I'd suggest investigating the properties to figure out which one best suits your needs.

To assume that the input errors are known entirely, use the guidelines from VarianceEstimatorFunction's documentation and set VarianceEstimatorFunction -> (1&), Weights -> 1/σy^2 in NonlinearModelFit, where 1/σy^2 is the inverse square of the expected error in y for each data point.